FINITE ELEMENT EXTERIOR CALCULUS FOR PARABOLIC EVOLUTION PROBLEMS ON RIEMANNIAN HYPERSURFACES

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1 FINITE ELEMENT EXTERIOR CALCULUS FOR PARABOLIC EVOLUTION PROBLEMS ON RIEMANNIAN HYPERSURFACES MICHAEL HOLST AND CHRIS TIEE ABSTRACT. Over te last ten years, te Finite Element Exterior Calculus (FEEC) as been developed as a general framework for linear mixed variational problems, teir numerical approximation by mixed metods, and teir error analysis. Te basic approac in FEEC, pioneered by Arnold, Falk, and Winter in two seminal articles in 2006 and 2010, interprets tese problems in te setting of Hilbert complexes, leading to a more general and complete understanding. Over te last five years, te FEEC framework as been extended to a broader set of problems. One suc extension, due to Holst and Stern in 2012, was to problems wit variational crimes, allowing for te analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Teir results substantially generalize te existing surface finite element approximation teory in several respects. In 2014, Gillette, Holst, and Zu extended te FEEC in anoter direction, namely to parabolic and yperbolic evolution systems by combining te FEEC framework for elliptic operators wit classical approaces for parabolic and yperbolic operators, by viewing solutions to te evolution problem as lying in Bocner spaces (spaces of Banac-space valued parametrized curves). Related work on developing an FEEC teory for parabolic evolution problems as also been done independently by Arnold and Cen. In tis article, we extend te work of Gillette-Holst-Zu and Arnold-Cen to evolution problems on Riemannian manifolds, troug te use of framework developed by Holst and Stern for analyzing variational crimes. We establis a priori error estimates tat reduce to te results from earlier work in te flat (non-criminal) setting. Some numerical examples are also presented. Date: September 21, Key words and prases. FEEC, elliptic equations, evolution equations, nonlinear equations, approximation teory, nonlinear approximation, inf-sup conditions, a priori estimates. MH was supported in part by NSF Awards , , and CT was supported in part by NSF Award

2 2 M. HOLST AND C. TIEE CONTENTS 1. Introduction Te Hodge eat equation and its mixed form Semidiscretization of te equation Error analysis Summary of te paper 5 2. Te Finite Element Exterior Calculus Hilbert Complexes Approximation of Hilbert Complexes Removing te Subcomplex Assumption: Variational Crimes Elliptic Error Estimates for a Nonzero Harmonic Part Abstract Evolution Problems Overview of Bocner Spaces and Abstract Evolution Problems Recasting te Problem as an Abstract Evolution Equation Error Estimates for te Abstract Parabolic Problem Discretization of te weak form Determining te error terms and teir evolution Parabolic Equations on Compact Manifolds Te de Ram Complex on a Manifold Approximation of a ypersurface in a tubular neigborood Finite element spaces Estimates for te Mixed Hodge Laplacian problem on manifolds A Numerical Example Conclusion and Future Directions 35 Appendix A. Acknowledgments 37 References INTRODUCTION Arnold, Falk, and Winter [2, 3] introduced te Finite Element Exterior Calculus (FEEC) as a general framework for linear mixed variational problems, teir numerical approximation by mixed metods, and teir error analysis. Tey recast tese problems using te ideas and tools of Hilbert complexes, leading to a more complete understanding. Subsequently, Holst and Stern [22] extended te Arnold Falk Winter framework to include variational crimes, allowing for te analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing te existing surface finite element approximation teory in several directions. Gillette, Holst, and Zu [20] extended te FEEC in anoter direction, namely to parabolic and yperbolic evolution systems by combining recent work on te FEEC for elliptic problems wit a classical approac of Tomée [33] to solving evolution problems using semi-discrete finite element metods, by viewing solutions to te evolution problem as lying in Bocner spaces (spaces of Banac-space valued parametrized curves). Arnold and Cen [1] independently developed related work, for generalized Hodge Laplacian parabolic problems for differential forms of arbitrary degree, and Holst, Mialik, and Szypowski [21] consider similar work in adaptive finite element metods. In tis article, we aim to combine te approaces of te above articles,

3 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 3 extending te work of Gillette, Holst, and Zu [20] and Arnold and Cen [1] to parabolic evolution problems on Riemannian manifolds by using te framework of Holst and Stern [22] Te Hodge eat equation and its mixed form. We now introduce our problem by some concrete motivation. We consider an evolution equation for differential forms on a manifold. Ten we reprase it as a mixed problem as an intermediate step toward semidiscretization using mixed finite element metods. We ten see ow tis allows us to leverage existing a priori error estimates for parabolic problems, and see ow it fits in te framework of Hilbert complexes. Let M be a compact oriented Riemannian n-manifold embedded in R n+1. Te Hodge eat equation is to find time-dependent k-form u : M [0, T ] Λ k (M) suc tat u t u = u t + (δd + dδ)u = f in M, for t > 0 u(, 0) = g in M. (1.1) were g is an initial k-form, and f, a possibly time-dependent k-form, is a source term. Note tat no boundary conditions are needed for manifolds witout boundary. Tis is te problem studied by Arnold and Cen [1], and in te case k = n, one of te problems studied by Gillette, Holst, and Zu [20], building upon work in special cases for domains in R 2 and R 3 by Jonson and Tomée [24, 33]. For te stability of te numerical approximations wit te metods of [23] and [3], we recast te problem in mixed form, converting te problem into a system of differential equations. Motivating te problem by setting σ = δu (recall tat for te Diriclet problem and k = n, δ ere corresponds to te gradient in Euclidean space, and is te adjoint d, corresponding to te negative divergence), and taking te adjoint, we ave σ, ω u, dω = 0, ω HΩ k 1 (M), t > 0, u t, ϕ + dσ, ϕ + du, dϕ = f, ϕ, ϕ HΩ k (M) t > 0. u(0) = g. (1.2) Unlike te elliptic case, we do not ave to explicitly account for armonic forms in te formulation of te equations temselves, but tey will definitely play a critical role in our analysis and bring new results not apparent in te k = n case Semidiscretization of te equation. In order to analyze te numerical approximation, we semidiscretize our problem in space. In our case, we sall assume, following [22], tat we ave a family of approximating surfaces M to te ypersurface M, given as te zero level set of some signed distance function, all contained in a tubular neigborood U of M, and a projection a : M M along te surface normal (of M). Te surfaces may be a triangulations, i.e., piecewise linear (studied by Dziuk and Demlow in [16, 14]), or piecewise polynomial (obtained by Lagrange interpolation over a triangulation of te projection a, as later studied by Demlow in [13]). We pull forms on M to M back via te inverse of te normal projection, wic furnises injective morpisms i k : Λk HΩk (M) as required by te teory in [22], wic we sall review in Section 2 below. Finally, we need a family of linear projections Π k : HΩk (M) Λ k suc tat Π i = id wic allow us to interpolate given data into te cosen finite element spaces tis is necessary because some of te more obvious, natural seeming coices of operators, suc as i, can be difficult to compute (neverteless, i will still be useful teoretically).

4 4 M. HOLST AND C. TIEE We now can formulate te semidiscrete problem: we seek a solution (σ, u ) H S HΩ k 1 HΩ k suc tat σ, ω u, dω = 0, ω H, t > 0 u,t, ϕ + dσ, ϕ + du, dϕ = Π f, ϕ, ϕ S t > 0 u (0) = g. (1.3) We sall describe ow to define g S sortly; it is to be some suitable interpolation of g. As S and H are finite-dimensional spaces, we can reduce tis to a system of ODEs in Euclidean space by coosing bases (ψ i ) for S and (φ k ) for H ; expanding te unknowns σ = i Σi (t)ψ i and u = k U k (t)φ k ; substituting tese basis functions as test functions to form matrices A kl = φ k, φ l, B ik = dψ i, φ k, and D ij = ψ i, ψ j ; and finally forming te vectors for te load data F defined by F k = F, φ k, and initial condition G defined by g = G k φ k. We tus arrive at te matrix equations for te unknown, time-dependent coefficient vectors Σ and U: DΣ B T U = 0, AU t + BΣ + KU = F, for t > 0 U(0) = G. Te matrices A and D are positive definite, ence invertible. Substituting Σ = D 1 B T U, we ave te system of ODEs AU t + (BD 1 B T + K)U = F, for t > 0, U(0) = G, wic as a unique solution by te usual ODE teory. For purposes of actually numerically integrating te ODE, namely, discretizing fully in space and time, it is better not to use te above formulation, because it can lead to dense matrices. Computationally, tis is due to te explicit presence of an inverse, D 1, not directly multiplying te variable; conceptually, tis is actually a statement about te discrete adjoint to te codifferential d generally aving global support even if te finite element functions are only locally supported [1]. Instead, we differentiate te first equation wit respect to time, getting DΣ t B T U t = 0, wic leads to te block system ( ) ( d D B T Σ = dt 0 A U) ( 0 0 B K) ( Σ U ) ( ) 0 + F (1.4) wic is still well-defined ODE for Σ and U, as te invertible matrices A and D appear on te diagonal. Tis differentiated equation also plays a role in te sowing tat te continuous problem is well-posed. Tese equations differ from tose studied by Gillette, Holst, and Zu [20], Arnold and Cen [1], and Tomée [33] by te coice of finite element spaces ere we are assuming tem to be in some Sobolev space of differential forms on manifolds (or in a triangulated mes in a tubular neigborood) rater tan subsets of Euclidean space. Tis suggests tat we sould try to gater tese commonalities, examine wat appens in abstract Hilbert complexes, and see ow general a form of error estimate we can get tis way Error analysis. Te general idea of te metod of Tomée [33] is to compare te semidiscrete solution to an elliptic projection of te data, a metod first explored by Weeler [35]. If we assume tat we already ave a solution u to te continuous problem, ten for eac fixed time t, u(t) can be considered as trivially solving an elliptic equation wit data u(t). Tus, using te metods developed in [3], we consider te discrete solution ũ for u in tis elliptic problem (namely, applying te discrete solution operator

5 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 5 T to u(t)). Tis may be compared to te true solution (at eac fixed time) using te error estimates in [3]. Wat remains is to compare te semidiscrete solution u (as defined by te ODEs (1.3) above) to te elliptic projection, so tat we ave te full error estimate by te triangle inequality. Tomée derives te following estimates, for finite elements in te plane (n = 2) of top-degree forms (k = 2, tere represented by a scalar proxy), for g te elliptic projection of te initial condition g and t 0: t ) u (t) u(t) L 2 c ( u(t) 2 H 2 + u t (s) H 2ds, (1.5) 0 ( t ) 1/2 ) σ (t) σ(t) L 2 c ( u(t) 2 H 3 + u t (s) 2 H 2ds. (1.6) Gillette, Holst, and Zu [20], and Arnold and Cen [1] generalize tese estimates and represent tem in terms of Bocner norms. Tese estimates describe te accumulation of error up to fixed time value t, assuming, of course, tat te spaces finite elements are sufficiently regular to allow tose estimates. Te key equation tat makes tese estimates possible are Tomée s error evolution equations: defining ρ = ũ (t) u(t), θ = u (t) ũ (t), and ε = σ (t) σ (t), we ave θ t, φ div ε(t), φ = ρ t, φ ε, ω + θ, div ω = 0. Tese are used to derive certain differential inequalities and make Grönwall-type estimates. In tis capter, we examine te above error equations and place tem in a more abstract framework. We use Bocner spaces (also used by [20]) to describe time evolution in Hilbert complexes, building on teir successful use in elliptic problems. We investigate Tomée s metod in tis framework to gain furter insigt into ow finite element error estimates evolve in time Summary of te paper. Te remainder of tis paper is structured as follows. In Section 2, we review te finite element exterior calculus (FEEC) and te variational crimes framework of Holst and Stern [22]. We prove some extensions in order to account for problems wit prescribed armonic forms; tis is wat allows te elliptic projection to work in te case were armonic forms are present. In Section 3, we formulate abstract parabolic problems in Bocner spaces and extend some standard results on te existence and uniqueness of strong solutions, and describe ow tis problem fits into tat framework. In Section 4, we extend te a priori error estimates for Galerkin mixed finite element metods to parabolic problems on Hilbert complexes. Ten, we relate te resuls to te problem on manifolds. Te main abstract result is Teorem 4.2, wic uses te previous results from te FEEC framework wit variational crimes, in order to understand ow tose error terms evolve wit time. We ten specialize, in Section 5 to parabolic equations on Riemannian manifolds, our original motivating example, and see ow tis generalizes te error estimates of Tomée [33], Gillette, Holst, and Zu [20], and Holst and Stern [22]. In Section 6, we present a numerical experiment comparing te metods based on tis mixed form to more straigtforward implementations in te scalar eat equation case. 2. THE FINITE ELEMENT EXTERIOR CALCULUS We review ere te relevant results from te finite element exterior calculus (FEEC) tat we will need for tis paper. FEEC was introduced in Arnold, Falk and Winter [2, 3] as a framework for deriving error estimates and formulating stable numerical metods for 0

6 6 M. HOLST AND C. TIEE a large class of elliptic PDE. One of te central ideas wic elped unify many of tese distinct metods into a structured framework as been te idea of Hilbert complexes [8], wic abstracts te essential features of te cocain complexes commonly found in exterior calculus and places tem in a context were modern metods of functional analysis may be applied. Tis assists in formulating and solving boundary value problems, in direct analogy to ow Sobolev spaces ave elped provide a framework for solving suc problems for functions. Arnold, Falk, and Winter [3] place numerical metods into tis framework by coosing certain finite-dimensional subspaces satisfying certain compatibility and approximation properties. Holst and Stern [22] extended tis framework by considering te case in wic tere is an injective morpism from a finite-dimensional complex to te complex of interest, witout it necessarily being inclusion. Tis allows te treatment of geometric variational crimes [6, 7], were an approximating manifold (on wic it may be far easier to coose finite element spaces) no longer coincides wit te actual manifold on wic we seek our solution. We review te teory as detailed in [22] and refer te reader tere for details Hilbert Complexes. As stated before, te essential details of differential complexes, suc as te de Ram complex, are nicely captured in te notion of Hilbert complexes. Tis enables us to see clearly were many elements of boundary value problems come from, in particular, te Laplacian, Hodge decomposition teorem, and Poincaré inequality. In addition, it allows us to see ow to carry tese notions over to numerical approximations. Definition 2.1 (Hilbert complexes). We define a Hilbert complex (W, d) to be sequence of Hilbert spaces W k wit possibly unbounded linear maps d k : V k W k V k+1 W k+1, suc tat eac d k as closed grap, densely defined, and satisfies te cocain property d k d k 1 = 0 (tis is often abbreviated d 2 = 0; we often omit te superscripts wen te context is clear). We call eac V k te domain of d k. We will often refer to elements of suc Hilbert spaces as forms, being motivated by te canonical example of te de Ram complex. Te Hilbert complex is called a closed complex if eac image space B k = d k 1 V k 1 (called te k-coboundaries is closed in W k, and a bounded complex if eac d k is in fact a bounded linear map. Te most common arrangement in wic one finds a bounded complex is by taking te sequence of domains V k, te same maps d k, but now wit te grap inner product v, w V = v, w + d k v, d k w. for all v, w V k. Unsubscripted inner products and norms will always be assumed to be te ones associated to W k. Definition 2.2 (Cocycles, Coboundaries, and Coomology). Te kernel of te map d k in V k will be called Z k, te k-cocycles and, as before, we ave B k = d k 1 V k 1. Since d k d k 1 = 0, we ave B k Z k, so we ave te k-coomology Z k /B k. Te armonic space H k is te ortogonal complement of B k in Z k. Tis means, in general, we ave an ortogonal decomposition Z k = B k H k, and we ave tat H k is isomorpic to Z k /B k, te reduced coomology, wic of course corresponds to te usual coomology for closed complexes. Definition 2.3 (Dual complexes and adjoints). For a Hilbert complex (W, d), we can form te dual complex (W, d ) wic consists of spaces Wk = W k, maps d k : V k Wk V k 1 W k 1 suc tat d k+1 = (dk ), te adjoint operator, tat is: d k+1v, w = v, d k w.

7 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 7 Te operators d decrease degree, so tis is a cain complex, rater tan a cocain complex; te analogous concepts to cocycles and coboundaries extend to tis case and we write Z k and B k for tem. Definition 2.4 (Morpisms of Hilbert complexes). Let (W, d) and (W, d ) be two Hilbert complexes. f : W W is called a morpism of Hilbert complexes if we ave a sequence of bounded linear maps f k : W k W k suc tat d k f k = f k+1 d k (tey commute wit te differentials). Wit te above, we can sow te following weak Hodge decomposition: Teorem 2.5 (Hodge Decomposition Teorem). Let (W, d) be a Hilbert complex wit domain complex (V, d). Ten we ave te W - and V -ortogonal decompositions were Z k V = Z k W V k. W k = B k H k Z k W (2.1) V k = B k H k Z k V. (2.2) Of course, if B k is closed, ten te extra closure is unnecessary, and we omit te term weak. We sall simply write Z k for Z k V, wic is will be te most useful ortogonal complement for our purposes. Te ortogonal projections P U for a subspace U will be in te W -inner product unless oterwise stated (altoug again, due to te two inner products coinciding on Z k and its subspaces, tey may be te same). We note tat by te abstract properties of adjoints ([3, 3.1.2]), Z k W = B k, and Bk W = Z k. Also very useful is tat te V - and W -norms agree on Z and ence on B and H. Te following inequality is an important result crucial to te stability of our solutions to te boundary value problems as well as te numerical approximations: Teorem 2.6 (Abstract Poincaré Inequality). If (V, d) is a closed, bounded Hilbert complex, ten tere exists a constant c P > 0 suc tat for all v Z k, v V c P d k v V. In te case tat (V, d) is te domain complex associated to a closed Hilbert complex (W, d), (V, d) is again closed, and te additional grap inner product term vanises: d k v V = d k v. We now introduce te abstract version of te Hodge Laplacian and te associated problem. Definition 2.7 (Abstract Hodge Laplacian problems). We consider te operator L = dd +d d on a Hilbert complex (W, d), called te abstract Hodge Laplacian. Its domain is D L = {u V k Vk : du V k+1, d u V k 1 }, and te Hodge Laplacian problem is to seek u V k V k, given f W k, suc tat du, dv + d u, d v = f, v (2.3) for all v V k Vk. Tis is simply te weak form of te Laplacian and any u V k Vk satisfying te above is called a weak solution. Owing to difficulties in te approximation teory for suc a problem (it is difficult to construct finite elements for te space V k Vk ), Arnold, Falk, and Winter [3] formulated te mixed abstract Hodge Laplacian problem by defining auxiliary variables σ = d u and p = P H f, te ortogonal projection of f into te armonic space, and considering a system of equations, to seek (σ, u, p) V k 1 V k H k suc tat σ, τ u, dτ = 0 τ V k 1 dσ, v + du, dv + p, v = f, v v V k u, q = 0 q H k. (2.4)

8 8 M. HOLST AND C. TIEE Te first equation is te weak form of σ = d u, te second is te main equation (2.3) modified to account for a armonic term so tat a solution exists, and te tird enforces uniqueness by requiring perpendicularity to te armonic space. Wit tese modifications, te problem is well-posed by considering te bilinear form (writing X k := V k 1 V k H k ) B : X k X k R defined by B(σ, u, p; τ, v, q) := σ, τ dτ, u + dσ, v + du, dv + p, v u, q. (2.5) and linear functional F (X k ) given by F (τ, v, q) = f, v. Te form B is not coercive, but rater, for a closed Hilbert complex, satisfies an inf-sup condition [3, 4]: tere exists γ > 0 (te stability constant) suc tat inf sup (σ,u,p) 0 (τ,v,q) 0 B(σ, u, p; τ, v, q) (σ, u, p) X (τ, v, q) X =: γ > 0. were we ave defined a standard norm on products: (σ, u, p) X := σ V + u V + p. Tis is sufficient to guarantee te well-posedness [4]. To summarize: Teorem 2.8 (Arnold, Falk, and Winter [3], Teorem 3.1). Te mixed variational problem (2.4) on a closed Hilbert complex (W, d) wit domain (V, d) is well-posed: te bilinear form B satisfies te inf-sup condition wit constant γ, so for any F (X k ), tere exists a unique solution (σ, u, p) to (2.4), i.e., B(σ, u, p; τ, v, q) = F (τ, v, q) fo all (τ, v, q) X k, and moreover, (σ, u, p) X γ 1 F X. Te stability constant γ 1 depends only on te Poincaré constant. Note tat te general teory (e.g., [4, 18]) guarantees a unique solution exists for any bounded linear functional F (X k ), wic in tis case wit product spaces, means tat te problem is still well-posed wen tere are oter nonzero linear functionals on te RHS of (2.4) besides f, v. We sall need tis result for parabolic problems, were we assume u as a armonic part (P H u 0) Approximation of Hilbert Complexes. We now approximate solutions to te abstract mixed Hodge Laplacian problem. To do so, Arnold, Falk, and Winter [3] introduce finite-dimensional subspaces V V of te domain complex, suc tat te inclusion i : V V is a morpism, i.e. dv k V k+1. Wit te weak form (2.4), we formulate te Galerkin metod by restricting to te subspaces: σ, τ u, dτ = 0 τ V k 1 dσ, v + du, dv + p, v = f, v v V k (2.6) u, q = 0 q H k. We abbreviate by setting X k k 1 := V V k Hk. We must also assume te existence of bounded, surjective, and idempotent (projection) morpisms π : V V. It is generally not te ortogonal projection, as tat fails to commute wit te differentials. As a projection, it gives te following quasi-optimality result: u π u V = inf v V (I π )(u v) V I π inf v V u v V. Te problem (2.6) is ten well-posed, wit a Poincaré constant given by c P π k, were c P is te Poincaré constant for te continuous problem. Tis guarantees all te previous abstract results apply to tis case. Wit tis, we ave te following error estimate:

9 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 9 Teorem 2.9 (Arnold, Falk, and Winter [3], Teorem 3.9). Let (V, d) be a family of subcomplexes of te domain (V, d) of a closed Hilbert complex, parametrized by and admitting uniformly V -bounded cocain projections π, and let (σ, u, p) X k be te solution of te continuous problem and (σ, u, p ) X k be te corresponding discrete solution. Ten te following error estimate olds: (σ σ, u u, p p ) X = σ σ V + u u V + p p C( inf τ V k 1 σ τ V + inf u v V + inf p q V + µ inf P B u v V ) v V k q V k v V k wit µ = µ k = sup ( r H I π) k r, te operator norm of I π k k restricted to H k. r =1 (2.7) Corollary If te V approximate V, tat is, for all u V, inf v V u v V 0 as 0, we ave convergence of te approximations. In general, te armonic spaces H k and H k do not coincide, but tey are isomorpic under many circumstances we sall consider (namely, te spaces are isomorpic if for all armonic forms q H k, te error q π q is at most te norm q itself [3, Teorem 3.4], and it always olds for te de Ram complex). For a quantitative estimate relating te two different kinds of armonic forms, we ave te following Teorem 2.11 ([3],Teorem 3.5). Let (V, d) be a bounded, closed Hilbert complex, (V, d) a Hilbert subcomplex, and π a bounded cocain projection. Ten (I P H )q V (I π k )q V, q H k (2.8) (I P H )q V (I π k )P H q V, q H k. (2.9) 2.3. Removing te Subcomplex Assumption: Variational Crimes. For geometric problems, it is essential to remove te requirement tat te approximating complex V actually be subspaces of V. Tis is motivated by te example of approximating planar domains wit curved boundaries by piecewise-linear approximations, resulting in finite element spaces tat lie in a different function space [6]. Holst and Stern [22] extend te Arnold, Falk, Winter [3] framework by supposing tat i : V V is an injective morpism wic is not necessarily inclusion; tey also require projection morpisms π : V V wit te property π i = id, wic replaces te idempotency requirement of te preceding case. To summarize, given (W, d) a Hilbert complex wit domain (V, d), (W, d ) anoter complex (wose inner product we denote, ) wit domain (V, d ), injective morpisms i : W W, and finally, projection morpisms π : V V. We ten ave te following generalized Galerkin problem: σ, τ u, d τ = 0 τ V k 1 d σ, v + d u, d v + p, v = f, v v V k (2.10) u, q = 0 q H k, were f is some interpolation of te given data f into te space W (we will discuss various coices of tis operator later). Tis gives us a bilinear form B (σ, u, p ; τ, v, q ) := σ, τ u, d τ + d σ, v + d u, d v + p, v u, q. (2.11) Tis problem is well-posed, wic again follows from te abstract teory as long as te complex is closed, and tere is a corresponding Poincaré inequality:

10 10 M. HOLST AND C. TIEE Teorem 2.12 (Holst and Stern [22], Teorem 3.5 and Corollary 3.6). Let (V, d) and (V, d ) be bounded closed Hilbert complexes, wit morpisms i : V V and π : V V suc tat π i = id. Ten for all v Z k, we ave v V c P π k i k+1 d v V, were c P is te Poincaré constant corresponding to te continuous problem. If (V, d) and (V, d ) are te domain complexes of closed complexes (W, d) and (W, d ), ten d v V is simply d v (since it is te grap norm and d 2 = 0). In oter words, te norm of te injective morpisms i also contributes to te stability constant for tis discrete problem. Analysis of tis metod results in two additional error terms (along wit now aving to explicitly reference te injective morpisms i wic may no longer be inclusions), due to te inner products in te space V no longer necessarily being te restriction of tat in V : te need to approximate te data f, and te failure of te morpisms i to be unitary. Teorem 2.13 (Holst and Stern [22], Corollary 3.11). Let (V, d) be te domain complex of a closed Hilbert complex (W, d), and (V, d ) te domain complex of (W, d ) wit morpisms i : W W and π : V V as above. Ten if we ave solutions (σ, u, p) and (σ, u, p ) to (2.4) and (2.10) respectively, te following error estimate olds: σ i σ V + u i u V + p i p C( inf τ i V k 1 σ τ V + were J = i i, and µ = µ k = sup ( I i k ) πk r. r H k r =1 inf v i V k u v V + inf q i V k p q V + µ inf v i V k P B u v V + f i f + I J f ), (2.12) Te extra terms (in te tird line of te inequality) are analogous te terms described in te Strang lemmas [6, III.1]. Te main idea of te proof of Teorem 2.13 (wic we will recall in more detail below, because we will need to prove a generalization of it as part of our main results) is to form an intermediate complex by pulling te inner products in te complex (W, d) back to (W, d ) back by i, construct a solution to te problem tere, and compare tat solution wit te solution we want. Tis modified inner product does not coincide wit te given one on W precisely wen i is not unitary: v, w i W = i v, i w = i i v, w = J v, w. Unitarity is simply te condition J = I. Te complex W wit te modified inner product now may be identified wit a true subcomplex of W, for wic te teory of [3] directly applies, yielding a solution (σ, u, p ) V k 1 V k H k, were H k is te discrete armonic space associated to te space wit te modified inner product. Tis generally does not coincide wit te discrete armonic space H k, since te discrete codifferential d in tat case is defined to be te adjoint wit respect to te modified inner product, yielding a different Hodge decomposition. Te estimate of i σ σ V + i u u V + i p p ten proceeds directly from te preceding teory for subcomplexes (2.7). Te variational crimes, on te oter and, arise from comparing te solution (σ, u, p ) wit (σ, u, p ). Finally, te error estimate (2.12) proceeds by te triangle inequality (and te boundedness of te morpisms i ).

11 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS Elliptic Error Estimates for a Nonzero Harmonic Part. Our objective in te remainder of tis section is to prove one of our main results, a generalization of Teorem 2.13 wic allows te possibility of te solution u aving a nonzero armonic part w. We first need a few lemmas. Lemma Teorem 2.9 continues to apply wen we ave u, q = w, q for all q H k, were w H k is prescribed (i.e., P H u = w, wic may generally not be zero). Proof. We closely follow te proof, in [3], of Teorem 2.9 above, noting were te modifications must occur. Let B be te bounded bilinear form (2.5); ten (σ, u, p) satisfies, for all (τ, v, q ) X k, B(σ, u, p; τ, v, q ) = f, v u, q. We V -ortogonally project (σ, u, p) in eac factor to (τ, v, q) X k. (τ, v, q ) X k, Ten for any B(σ τ, u v, p q; τ, v, q ) = B(σ τ, u v, p q; τ, v, q ) + u, q w, q = B(σ τ, u v, p q; τ, v, q ) + P H (u w), q C ( σ τ V + u v V + p q + P H (u w) ) ( τ V + v V + q ). (2.13) Noticing tat te factor p q in te bilinear form above is in te original domain H k, we can now coose te appropriate (τ, v, q ) tat verifies inf-sup condition of B: B(σ τ, u v, p q; τ, v, q ) γ( σ τ V + u v V + p q )( τ V + v V + q ). Comparing tis to (2.13) above, we may cancel te common factor, and divide by γ to arrive at σ τ V + u v V + p q Cγ 1 ( σ τ V + u v V + p q + P H (u w) ). (2.14) Tis differs (aside from te notation) from [3] in tat we now ave, rater tan P H u, instead P H (u w), wit te armonic part subtracted off. Removing te armonic part allows us to continue as in [3]: te Hodge decomposition u w = u P H u consists only of coboundary and perpendicular terms u B + u B k Z k. Wit H k contained in Z k, it follows P H u = 0, and P H π u B = 0. Also, (I π )u B is perpendicular to H k. Terefore, for all q H k, P H (u P H u), q = P H u B, q = P H (u B π u B ), q Now, setting we ave = u B π u B, q = u B π u B, (I P H )q. q = P H (u P H u) P H (u P H u) Hk, P H (u P H u) = P H (u P H u), q = u B π u B, (I P H )q u B π u B (I P H )q C (I P H )q inf u B v V. v V k

12 12 M. HOLST AND C. TIEE Finally, by te second estimate of Teorem 2.11 above, we can bound (I P H )q by (I π )P H q, giving us (I P H )q (I π )P H q sup r =1 r H k (I π )r P H q µ. From te triangle inequality, we derive te estimate σ σ V + u u V + p p σ τ V + u v V + p q + τ σ V + u v V + q p ( ) (1 + Cγ 1 ) σ τ V + u v V + p q + µ inf P B u v V. v i V k Using best approximation property of ortogonal projections, we can express te remaining terms wit te infima, and tis gives te result. We also need a tecnical lemma wic enables us to identify te ortogonal projection onto te identified subcomplex i X k in order to be able to make additional estimates of te variational crimes in terms of te operator norms I J. It is te infinitedimensional analogue of taking a Moore-Penrose pseudoinverse [32, 3.3] for infinitedimensional spaces: Lemma Let i : W W be an injective map of Hilbert spaces, and J = i i. Ten J is invertible, and J 1 i is te Moore-Penrose pseudoinverse of i, i.e. it maps i W isometrically back to W wit te modified inner product. We write i + for J 1 i. Proof. Te invertibility of J follows directly from te injectivity of i, wic makes J, a positive-definite form. Now, (J 1 i )i = J 1 J = id W, wic sows tat it is in fact a left inverse, as required for pseudoinverses. To sow te ortogonality, minimizing 1 i 2 w b 2 for any b W yields, by te completeness of W, te solution w = J 1 i b, sowing tat it is a least squares solution, terefore te Moore-Penrose pseudoinverse. We are now ready to prove our main elliptic error estimate, an extension of Teorem Teorem 2.16 (Extension of elliptic error estimates to allow for a armonic part). Consider te problems (2.4) and (2.10) but instead wit now wit prescribed, possibly nonzero armonic part w: Given f W k and w H k, we seek (σ, u, p) X k suc tat σ, τ u, dτ = 0 τ V k 1 dσ, v + du, dv + p, v = f, v v V k u, q = w, q q H k. (2.15) Te solution to tis problem exists and is unique, wit w indeed equal to P H u, and is bounded by c( f + w ), wit c depending only on te Poincaré constant. Now, consider te discrete problem, wit f, w V k: σ, τ u, d τ = 0 τ V k 1 d σ, v + d u, d v + p, v = f, v v V k (2.16) u, q = w, q q H k.

13 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 13 Tis problem is also well-posed, wit te modified Poincaré constant in Teorem Ten we ave te following generalization of te error estimate (2.12) above: σ i σ V + u i u V + p i p ( C inf τ i V k 1 σ τ V + inf u v V + inf v i V k q i V k p q V + µ inf v i V k P B u v V + inf w ξ V + f i f + w i w + I J ( f + w ) ξ i V k were, as before, J = i i, and µ = µ k = sup ( I i k ) πk r. r H k r =1 ), (2.17) We see tat tree new error terms arise from te approximation of te armonic part, one being te data interpolation error (but measured in te V -norm, wic partially captures ow d fails to commute wit i and ow w may not necessarily be a discrete armonic form), anoter best approximation term, and finally anoter term from te nonunitarity. Te relation of f to f and w to w need not be furter specified, because te teorem directly expresses suc a dependence in terms of teir relation to i f and i w; it as been isolated as a separate issue. However as mentioned in te introduction, and following [22], we often take f = Π f, were Π is some family of linear interpolation operators wit Π i = id. Anoter seemingly obvious coice is i itself (tus making tose corresponding error terms zero), but as mentioned in [22], tis can be difficult to compute, so we do not restrict ourselves to tis case. Various coices of interpolation will be crucial in deciding wic estimates to make in te parabolic problem. We split te proof of tis teorem into two parts, te first of wic derives te quantities on te second line of (2.17), and te second part, we derive te quantities on te tird line of (2.17). Generally, we follow te pattern of proof in [3, Teorem 3.9] and [22, Teorem 3.10], noting te necessary modifications, as well as a similar tecnique given for te improved error estimates by Arnold and Cen [1]. First part of te proof of Teorem First, following Holst and Stern [22] as above, we construct te complex W but wit te modified inner product J, (te associated domain complex V remains te same). Tis gives us a discrete Hodge decomposition wit anoter type of ortogonality and corresponding discrete armonic forms and ortogonal complement (due to a different adjoint d ): V k = B k H k Z k (generally, primed objects will represent te corresponding objects defined wit te modified inner product; te discrete coboundaries are in fact te same as before, because d and d do not depend on te coice of inner product). Te main complications arise in aving to keeping careful track of te different armonic forms involved, because teir nonequivalence and possible non-preservation by te operators contribute directly to te error. We ten define, as in [22], te intermediate solution (σ, u, p ) V k 1 (wic we abbreviate as X k ): V k H k J σ, τ J u, d τ = 0 τ V k 1 J d σ, v + J d u, d v + J p, v = i f, v v V k (2.18) J u, q = i w, q q H k,

14 14 M. HOLST AND C. TIEE and te corresponding bilinear form B : X X R given by B (σ, u, p ; τ, v, q ) := J σ, τ J u, d τ + J d σ, v + J d u, d v + J p, v J u, q. (2.19) Tis satisfies te inf-sup condition wit Poincaré constant c P π. Note tat we will need to extend all te bilinear forms B, and B k in te last factor to all of V in order to compare te two, since tey are initially only defined on te respective, differing armonic form spaces. Tis is not a problem so long as we remember to invoke te infsup condition only wen using te non-extended versions. Te idea is, again, to use te triangle inequality: σ i σ V + τ i τ V + p i p (2.20) σ i σ V + τ i τ V + p i p (2.21) + i (σ σ ) V + i (τ τ ) V + i (p p ). (2.22) Tese quantities can be estimated using only geometric properties of te domain; we ave no need to actually explicitly compute (σ, u, p ). To estimate te term (2.21) (wic we sall refer to as te PDE approximation term, wereas (2.22) will be called variational crimes), we recall tat i is an isometry of W wit te modified inner product to its image, wic is a subcomplex. Tus, Lemma 2.14 above applies, wit te approximation (i σ, i u, i p ) on identified subcomplex i X k. Tis gives us te terms on te second line of (2.17). To finis our main proof, we need to consider te variational crimes (2.22). Since te maps i are bounded, and we eventually absorb teir norms into te constant C above, it suffices to consider σ σ V + u u V + p p, wic we sall state as a separate teorem. Teorem Let (σ, u, p ) X k be a solution to (2.16), (σ, u, p ) X k a solution to (2.18), and w = P H u, te prescribed armonic part of te continuous problem. Ten σ σ V + u u V + p p C( f i f + w i w V + I J ( f + w ) + inf ξ i V k w ξ V ), (2.23) i.e., tey are bounded by te terms on te tird line in (2.17). Proof of Teorem 2.17 and second part of te proof of Teorem We follow te proof of Holst and Stern [22, Teorem 3.10] and note te modifications. Let (τ, v, q) and (τ, v, q ) X k. Consider te bilinear form B, (2.11) above, and write B (σ τ, u v, p q; τ, v, w ) = B (σ σ, u u, p p ; τ, v, q ) + B (σ τ, u v, p q; τ, v, q ).

15 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 15 We ten ave, recalling te modified bilinear form B, (2.19) above, and extending it in te last factors to all of V k, B (σ, u, p ; τ, v, q ) = B (σ, u, p ; τ, v, q ) + (I J )σ, τ (I J )u, d τ + (I J )d σ, v + (I J )d u, d v + (I J )p, v (I J )u, q. Substituting te respective solutions (2.16) and (2.18) (and noting te sligt discrepancy in te use of different armonic forms), we ave so B (σ, u, p ; τ, v, q ) = i f, v J u, q B (σ, u, p ; τ, v, q ) = f, v w, q, B (σ σ, u u, p p ; τ, v, q ) = f i f, v + u, q w, q (I J )σ, τ + (I J )u, d τ (I J )d σ, v (I J )d u, d v (I J )p, v. As before, we bound te form above and below. For te upper bound, using Caucy- Scwarz to estimate te extra inner product terms, we arrive at B (σ τ, u v, p q; τ, v, q ) C ( f i f + P H (u w ) + I J ( σ V + u V + p ) + σ τ V + u v V + p q ) ( τ V + v V + q ). For te lower bound, we again coose (τ, σ, q ) X k to verify te inf-sup condition tis time for B : B (σ τ, u v, p q; τ, v, q ) γ ( σ τ V + u v V + p q ) ( τ V + v V + q ) and γ depends only on te Poincaré constant c P i π, uniformly bounded in. Comparing wit te upper bound and dividing out te common factor as before, tis leads to: σ τ V + u v V + p q Cγ 1 ( f i f + P H (u w ) + I J ( σ V + u V + p ) + σ τ V + u v V + p q ). Coosing (τ, v, q) = (σ, u, P H p ), applying te triangle inequality wit p to account for te mismatc in te armonic spaces, and using te well-posedness of te continuous problem (2.18), σ σ V + u u V + p p C ( f i f + P H (u w ) + I J ( f + w ) + p q ). Tis differs from [22] in tat we ave te bound in terms of f + w, and tat we must estimate P H (u w ) rater tan P H u alone. First, we use te modified Hodge

16 16 M. HOLST AND C. TIEE decomposition to uniquely write u as u B + P H u + u wit u B Bk and u Zk, and P H (u w ) P H (u B + u ) + P H (P H u w ). (Te projection P H is respect to te modified inner product). For te first term, we proceed exactly as in [22]: we ave P H u B = 0 since te coboundary space is still te same, and tus only te term u contributes. Now u Zk so, using J to express it in terms of V -ortogonality, we ave J u Zk, and tus P H J u = 0. Terefore, we ave P H (u B + u ) = P H u = P H (I J )u C I J ( f + w ). For te p term, tis also proceeds as in [22] uncanged (except for, of course, te extra w term): using te (unmodified) discrete Hodge decomposition, we ave p = P B p + P H p = P B p + q. Since p H k, a similar argument gives J p Bk, so P B J p = 0 and p q = P B p = P B (I J )p C I J ( f + w ). Finally, we must consider te term P H (P H u w ). Expressing u in terms of w, te terms do not combine as easily as te analogous terms involving f and i f, because teir action as linear functionals operate on different armonic spaces. Continuing wit te proof of te teorem, we recall te tird equation of (2.18): J u, q = i w, q = J (J 1 i w), q wic terefore says P H u = P H i+ w. Tis enables us to properly work wit te modified ortogonal projection P H. Because i + is an isometry of te subspace i W to W, we ave P H i + w = i+ P i H w. were now P i H is te ortogonal projection onto te identified image armonic space. Ten, using te triangle inequality again, P H (P H u w ) ( PH PH i + w i+ w) + P H (J 1 i w i w) + P H (i w w ) P H ( i + ( ) I P i H w + J 1 I J ) i w + i w w C ( ( ) I P i H ) w + I J w + i w w. Te last term is te data approximation error for w, and te second term combines wit te previous errors tat reflect te non-unitarity of te operator. So, all tat remains is to estimate te first term. Since it is in te subcomplex i W, te first estimate of Teorem 2.11 applies: ( I P i H ) w (I π )w C inf w ξ V, (2.24) ξ i V k by quasi-optimality. Concluding remarks of te proof of Teorem To summarize, we proved our Main Teorem 2.16 by defining an intermediate solution on a modified complex tat we identify wit a subcomplex, and analyzing te result via te Arnold, Falk, and Winter [3] framework. Tat teorem olds, wit te estimate uncanged, toug now u and u no longer are perpendicular to teir respective armonic spaces. Te place were te extra terms all sow up is in te variational crimes. In te process of arriving at a term tat

17 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 17 looks like i w w, working wit te different armonic forms produces two more nonunitarity terms I J ( f + w ), and finally, using Teorem 2.11 yields a direct estimate of ow w fails to be a modified discrete armonic form, giving te last best approximation term inf ξ i V k w ξ V. We also note for future reference tat in spaces were we ave improved error estimates (wic means π are W -bounded maps) tat we can replace tat last V -norm in (2.24) to be te W -inner product. Finally, we remark tat, for a certain types of data interpolation, te errors f i f and w i w can be rewritten in terms of te oter errors and anoter best approximation term. Tis will be useful for us in our examples. Teorem 2.18 (Holst and Stern [22], Teorem 3.12). If Π : W k W k is a family of linear projections uniformly bounded wit respect to, ten for all f W k, ( ) Π f i f C I J f + inf φ i W k f φ. (2.25) 3. ABSTRACT EVOLUTION PROBLEMS In order to solve and approximate linear evolution problems, we introduce te framework of Bocner spaces (also following Gillette, Holst, and Zu [20]), wic realizes time-dependent functions as curves in Banac spaces (wic will correspond to spaces of spatially-dependent functions in our problem). We follow mostly [30] and [18] for tis material Overview of Bocner Spaces and Abstract Evolution Problems. Let X be a Banac space and I := [0, T ] an interval in R wit T > 0. We define C(I, X) := {u : I X u bounded and continuous}. In analogy to spaces of continuous, real-valued functions, we define a supremum norm on C(I, X), making C(I, X) into a Banac space: u C(I,X) := sup u(t) X. t I We will of course need to deal wit norms oter tan te supremum norm, wic motivates us to define Bocner spaces: to define L p (I, X), we complete C(I, X) wit te norm ( ) 1/p u L p (I,X) := u(t) p X dt. Similarly, we ave te space H 1 (I, X), te completion of C 1 (I, X) wit te norm ( 2 1/2 u H 1 (I,X) := u(t) 2 X + d dt u(t) dt). I Tere are metods of formulating tis in a more measure-teoretic way ([18, Appendix E]), considering Lebesgue-measurable subsets of I. As mentioned before, for our purposes, X will be some space of spatially-dependent functions, and te time-dependence is captured as being a curve in tis function space (altoug tis interpretation is only correct wen we are considering C(I, X) we must be careful about evaluating our functions at single points in time witout an enclosing integral). Usually, X will be a space in some Hilbert complex, suc as L 2 Ω k (M) or H s Ω k (M) were te forms are defined over a Riemannian manifold M. I X

18 18 M. HOLST AND C. TIEE We introduce tis framework in order to be able to formulate parabolic problems more generally. It turns out to be useful to consider te concept of rigged Hilbert space or Gelfand triple, wic consists of a triple of separable Banac spaces V H V suc tat V is continuously and densely embedded in H. For example, if (V, d) is te domain complex of some Hilbert complex (W, d), setting V = V k and H = W k works, as well as various combinations of teir products (so tat we can use mixed formulations). H is also continuously embedded in V. Te standard isomorpism (given by te Riesz representation teorem) between V and V, is not generally te composition of te inclusions, because te primary inner product of importance for weak formulations is te H-inner product. It coincides wit te notion of distributions acting on test functions. Writing, for te inner product on H, te setup is designed so tat wen it appens tat some F V is actually in H, we ave F (v) = F, v (wic is wy we will often write F, v to denote te action of F on v even if F is not in H). In fact, in most cases of interest, te H-inner product is te restriction of a more general bilinear form between two spaces, in wic elements of te left (acting) space are of less regularity tan elements of H, wile elements of te rigt space ave more regularity. Given A C(I, L(V, V )), a time-dependent linear operator, we define te bilinear form a(t, u, v) := A(t)u, v, (3.1) for (t, u, v) R V V. To proceed, as in elliptic problems, we need a to satisfy some kind of coercivity condition, altoug it need not be as strong. It turns out tat Gårding s Inequality is te rigt condition to use ere: a(t, u, u) c 1 u 2 V c 2 u 2 H, (3.2) wit c 1, c 2 constants independent of t I. Ten te following problem is te abstract version of linear, parabolic problems: Tis problem is well-posed: u t = A(t)u + f(t) (3.3) u(0) = u 0. (3.4) Teorem 3.1 (Existence of Unique Solution to te Abstract Parabolic Problem, [30], Teorem 11.3). Let f L 2 (I, V ) and u 0 H, and a te time-dependent quadratic form in (3.1). Suppose (3.2) olds. Ten te abstract parabolic problem (3.3) as a unique solution u L 2 (I, V ) H 1 (I, V ). Moreover, te Sobolev embedding teorem implies u C(I, H), wic allows us to unambiguously evaluate te solution at time zero, so te initial condition makes sense, and te solution indeed satisfies it: u(0) = u 0. Tis teorem is proved via standard metods ([30, p. 382]); we take an ortonormal basis of H tat is simultaneously ortogonal for V (a frequent situation occurring wen, say, it is an ortonormal basis of eigenfunctions of te Laplace operator), formulate te problem in te finite-dimensional subspaces, and use a priori bounds on suc solutions to extract a weakly convergent subsequence. Wit tis framework, we can sow tat a wide class of PDE problems, particularly ones tat are suited to finite element approximations, are well-posed.

19 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS Recasting te Problem as an Abstract Evolution Equation. Let us now see ow tese results apply in te case of te Hodge eat equation (1.1) on manifolds. We take a sligtly different approac from wat is done in [20] and [1], solving an equivalent problem. Tis sets tings up for our modified numerical metod detailed in later sections. Let (W, d) be a closed Hilbert complex, wit domain complex (V, d), te standard setup in te above in particular, we ave te Poincaré inequality and te well-posedness of te continuous Hodge Laplacian problem. We consider te space Y k := V k 1 V k and its dual Y = (V k 1 ) (V k ) wit te obvious product norms (we use primes to denote dual spaces so as not to conflict wit te dual complex wit respect to te Hodge star defined earlier, toug tese uses are related). Tis, along wit H = W k 1 W k, gives rigged Hilbert space structure Y H Y. Te embeddings are dense and continuous by definition of te grap inner product and tat te operators d ave dense domain. We consider te Bocner mixed weak parabolic problem: to seek a weak solution (u, σ) L 2 (I, Y) H 1 (I, Y ) to te mixed problem σ, ω u, dω = 0, ω V k 1, t I, u t, ϕ + du, dϕ + dσ, ϕ = f, ϕ, ϕ V k, t I, u(0) = g, (3.5) tis makes it suitable for approximation using finite-dimensional subspaces of Y (e.g. degrees of freedom for finite element spaces). We see tat (3.5) is te mixed form of (1.1), wic amounts to defining a system of differential equations, introducing te variable σ defined by σ = d u, were d is te adjoint of te operator d. We write te equation weakly (namely, moving d back to te oter side), wic makes no difference at te continuous level, but will make a significant difference wen discretizing. In order to use te abstract macinery above, we need a term wit σ t. Formally differentiating te first equation of (1.2), and substituting ϕ = dω in te second equation, we obtain 0 = σ t, ω u t, dω = σ t, ω f, dω + dσ, dω + du, ddω. Since d 2 = 0, tat last term vanises, and so, togeter wit te equation for u t, we ave te following system: σ t, ω + dσ, dω = f, dω, ω V k 1, t I, u t, ϕ + dσ, ϕ + du, dϕ = f, ϕ, ϕ V k, t I, u(0) = g. (3.6) Teorem 3.2. Suppose te initial condition g is in te domain of te adjoint V and f L 2 (I, (V k ) ). Ten te problem (3.6) is well-posed: tere exists a unique solution (σ, u) L 2 (I, Y) H 1 (I, Y ) C(I, H) wit (σ(0), u(0)) = (d g, g). Proof. We see tat given f L 2 (I, (V k ) ), we ave tat te functional F : (τ, v) f, dτ + f, v is in L 2 (I, Y ), since d maps V k 1 to V k. For an initial condition on σ, we can demand tat σ(0) be te unique σ 0 statisfying σ 0, τ g, dτ = 0. For tis to reasonably old, we must actually ave at least u 0 Vk, te domain of te adjoint operator d, tat is, σ 0 = d g. We equip te spaces wit te standard inner products for

20 20 M. HOLST AND C. TIEE product spaces: Consider te operator A : Y Y defined by (σ, u), (τ, v) H := σ, τ + u, v (3.7) (σ, u), (τ, v) Y := σ, τ V + u, v V. (3.8) a(σ, u; ω, ϕ) = A(σ, u), (ω, ϕ) = dσ, dω + dσ, ϕ + du, dϕ. Wit te functional F defined as above, we ave F L 2 (I, Y ), and so (3.6) is equivalent to te problem (σ, u) t = A(σ, u) + F. (3.9) We now need to verify tat te bilinear form a satisfies Gårding s Inequality: a(σ, u; σ, u) = dσ 2 + dσ, u + du 2 = σ 2 V σ 2 + dσ, u + u 2 V u 2 σ 2 V σ 2 dσ u + u 2 V u 2 σ 2 V σ σ 2 V 1 2 u 2 V + u 2 V u 2 = 1 2 (σ, u) 2 Y (σ, u) 2 H. Tus, te abstract teory applies, and noting tat te initial conditions (d g, g) H, we ave tat (σ, u) L 2 (I, Y) H 1 (I, Y ) C(I, H) is te unique solution to (3.6) wit initial conditions given by u(0) = g Vk and σ(0) = d g. Given tis, owever, we must still establis tat we also ave a solution to te original mixed problem (wic will be crucial in our error estimates): Teorem 3.3. Let (σ, u) L 2 (I, Y) H 1 (I, Y ) C(I, H) solve (3.6) above wit te initial conditions. Ten, in fact, (σ, u) also solves (3.5). Proof. Te second equation already olds, as it is incorporated uncanged into te equations (3.6). To sow te first equation, we sow σ t, ω u t, dω = 0 for all time t. Ten, since te original mixed equation olds at te initial time, standard uniqueness ensures it olds for all t I. We simply realize it is setting ϕ = dω: σ t, ω u t, dω = (σ, u) t, (ω, dω) H = a(σ t, u t ; ω, dω) + f, dω + f, dω = dσ, dω + dσ, dω + du, ddω = ERROR ESTIMATES FOR THE ABSTRACT PARABOLIC PROBLEM We now combine all te preceding abstract teory (te Holst-Stern [22] framework recalled in 2.2, and te abstract evolution problems framework recalled in 3) to extend te error estimates of Gillette, Holst, and Zu [20] and in particular, recover te case of approximating parabolic equations on compact, oriented 1 Riemannian ypersurfaces in 1 Using differential pseudoforms ([19, 2.8], [34]), we can eliminate tis restriction. However, more teory needs to be developed for tat case; te normal projection, in particular. We consider tis in future work.

21 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 21 V k (it is Xk missing te armonic factor), i : V V injective morpisms (tat are W -bounded), π : V V projection morpisms (wic may be merely V -bounded), and π i = id. Finally, we consider data interpolation operators Π : W W, suc tat Π i = id tat realize wic projections for te inomogeneous and prescribed armonic terms (f and w in te abstract teory above) tat we use. R n+1 wit triangulations in a tubular neigborood. Te key equation in te derivation of te estimates are te generalizations of Tomée s evolution equations for te error terms. We sall see tat tese equations lead most naturally to te use of certain Bocner norms for te error estimates tat are different for eac component in te equation. Let (W, d) be a closed Hilbert complex wit domain (V, d), and te Gelfand triple Y H Y on tis complex as above. Now consider our previous standard setup of finite-dimensional approximating complexes (W, d) wit domain (V, d), wit corresponding spaces Y k = V k Discretization of te weak form. Suppose we ave f L 2 (I, (V k ) ) and g V k. Let (σ, u) L 2 (I, Y) H 1 (I, Y ) C(I, H) be te unique (continuous) solution to (3.5), as covered in 3. As in [20], we can consider approximations to tis solution as functionals on finite-dimensional spaces Y, e.g. finite element spaces. Wit te above considerations, we formulate te semi-discrete Bocner parabolic problem: Find (σ, u ) : I Y suc tat σ, ω u, dω = 0, ω V k 1, t I u,t, ϕ + dσ, ϕ + du, dϕ = Π f, ϕ, ϕ V k, t I u (0) = g. (4.1) (We use te notation of Tomée for te test forms.) We define g, te projected initial data, sortly. A similar argument as in 3 above, differentiating te first equation wit respect to time, considering te Gelfand triple Y k W k 1 W k (Yk ) gives tat tis problem is well-posed (or more simply, we coose bases and reduce to standard ODE teory as in (1.3) above). Following Gillette, Holst, and Zu [20], we define te timeignorant discrete problem, using te idea of elliptic projection [35] wic we use to define a discrete solution via elliptic projection of te continuous solution at eac time t 0 I: We seek ( σ, ũ, p ) X k suc tat σ, ω ũ, dω = 0, ω V k 1 d σ, ϕ + dũ, dϕ + p, ϕ = Π ( u(t 0 )), ϕ, ϕ V k ũ, q = Π (P H u(t 0 )), q q H k. (4.2) Note tat we ave included a prescribed armonic form given by te armonic part of u (following [1]). We ten take te initial data g to be ũ (0); it is just te solution to te elliptic problem wit load data Π ( g), since u(0) = g. Note we do not directly interpolate g itself via Π for te data; te reason for tis will be seen sortly. Tis discrete problem is well-posed, i.e., a unique solution u (t 0 ) always exists for every time t 0 I, by te first part of Teorem 2.16 above. Te presence of an additional term p and equation involving armonic forms departs from Gillette, Holst, and Zu [20], because te teory tere is facilitated by te fact tat tere are no armonic n-forms on open domains in R n (te natural boundary conditions for suc spaces are Diriclet boundary conditions, in contrast to te more classical example of 0-forms, i.e. functions). Here, owever, we must consider armonic forms, since we may not be working at te

22 22 M. HOLST AND C. TIEE end of an abstract Hilbert complex. For our model problem, namely differential forms on compact orientable manifolds (witout boundary), even in te case of n-forms, te teory is completely symmetric (by Poincaré duality [5, 25, 28]). 2 In addition, te linear projections Π may not preserve te armonic space, wic gives te possibility of a nonzero p, despite u aving zero armonic part (so it is its own error term) Determining te error terms and teir evolution. Continuing te metod of Tomée [33], we use te time-ignorant discrete solution as an intermediate reference, and estimate te total errors by comparing to tis reference and using te triangle inequality. Rougly speaking, we try to estimate as follows: i σ (t) σ(t) V i σ (t) i σ (t) V + i σ (t) σ(t) V (4.3) i u (t) u(t) V i u (t) i ũ (t) V + i ũ (t) u(t) V. (4.4) It turns out tat tis grouping of te terms is not te most natural for our purposes. We sall see it is te structure of te error evolution equations tat groups te terms more naturally as: i u (t) u(t) (4.5) i σ (t) σ(t) + d(i u (t) u(t)) (4.6) d(i σ (t) σ(t)). (4.7) Te sum of tese tree terms is te sum of te two V -norms above. In addition, we sall see in our application to ypersurfaces tat tis particular grouping of te error terms also corresponds more precisely to te order of approximations in te improved estimates for te elliptic projection (namely, tey are of orders r+1, r, and r 1, respectively, for degree-r polynomial differential forms). Te plan is to use te teory of Holst and Stern [22] reviewed in 2.2 above to estimate te sum of te two second terms in (4.3) and (4.4); te elliptic projection simply is an approximation, at eac fixed time, of te trivial case of u being te solution of te continuous problem wit data given by its own Laplacian, u. Te armonic form portion will come up naturally as part of te calculuation. Using te notation of Tomée [33], we define te error functions ρ(t) := ũ (t) i u(t) (4.8) θ(t) := u (t) ũ (t) (4.9) ψ(t) = σ (t) i σ(t). (4.10) ε(t) := σ (t) σ (t) (4.11) (Tomée does not define te tird term ψ; we ave added it for convenience.) In te case tat tere are no variational crimes (i.e., J is unitary), te error terms ρ and ψ are bounded above by te elliptic projection errors (because tere, i is te ortogonal projection, and i = i = 1), so tat we ave, for example, tat i u u θ + ρ, corresponding to te use of ρ in [33, 20]. For our purposes, owever, te coice of ρ ere does not correspond as neatly, now being an intermediate quantity tat elps us estimate θ in terms te elliptic projection error (te second term in (4.4)). We find tat it contributes more terms wit I J. Similar remarks apply for σ and ψ. We use te metod of Tomée to estimate te terms θ and ε in terms of (te time derivatives 2 Despite tis, tere are a number of reasons wy one sould still prefer to continue to prase problems in terms of n-forms if te problem calls for it ([19] describes ow it affects te interpretation of certain quantities); and we sall see tat it does in fact still make a difference at te discrete level.

23 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 23 of) ρ and ψ, and te elliptic projection error; In order to do tis, we need an analogue of Tomée s error equations. Lemma 4.1 (Generalized Tomée error equations). Let θ, ρ, and ε be defined as above. Ten for all t I, ε, ω θ, dω = 0 ϕ V k 1, θ t, ϕ + dε, ϕ + dθ, dϕ = ρ t + p + (Π i )u t, ϕ ω V k. (4.12) Tis differs from Tomée [33] and Gillette, Holst, and Zu [20] wit te armonic term p, wic accounts for te projections Π possibly not sending te armonic forms to te discrete armonic forms, an extra dθ term wic accounts for possibly working away from te end of te complex (for differential forms on an n-manifold, forms of degree k < n), and anoter data interpolation error term for u t (wic also distinguises it from Arnold and Cen [1]). Proof. Te first equation is simply weakly expressing ε as d θ. Tis follows immediately from te corresponding equations in te semidiscrete problem and te time-ignorant discrete problem. For te second term, consider te expression B := θ t, ϕ + dε, ϕ + dθ, dϕ + ρ t, ϕ, (4.13) and expand it using te definitions to obtain B = u,t, ϕ ũ,t, ϕ + dσ d σ, ϕ + du dũ, dϕ + ũ,t, ϕ i u t, ϕ. We cancel te ũ,t terms, and apply te semidiscrete equation (4.1) to cancel te dσ and du terms, wic gives us B = Π f, ϕ d σ, ϕ dũ, dϕ i u t, ϕ, and finally, using te second equation of (4.2) to account for te middle terms, we ave B = Π f, ϕ + Π ( u), ϕ + p, ϕ i u t, ϕ = Π ( u + f u t ), ϕ + p, ϕ + (Π i )u t, ϕ. But since u t = u + f is te strong form of te equation, wic we know is satisfied by te uniqueness, it follows tat B = p + (Π i )u t, ϕ. Subtracting te ρ t from bot sides gives te result. Now we present our main teorem. Teorem 4.2 (Main parabolic error estimates). Let (σ, u) be te solution to te continuous problem (3.5), (σ, u ) be te semidiscrete solution (4.1), ( σ, ũ ) te elliptic projection (4.2), and te error quantities (4.8)-(4.11) be defined as above. Ten we ave te following error estimates: θ(t) ρ t L 1 (I,W ) + p L 1 (I,W ) + (Π i )u t L 1 (I,W ) (4.14) dθ(t) + ε(t) C ( ) ρ t L 2 (I,W ) + p L 2 (I,W ) + (Π i )u t L 2 (I,W ) (4.15) dε(t) C ( ψ t L 2 (I,W ) + d (Π i )u t L 2 (I,W )), (4.16)

24 24 M. HOLST AND C. TIEE wit ρ t L 2 (I,W ) C ( ) i ũ,t u t L 2 (I,W ) + I J L(W ) u t L 2 (I,W ) (4.17) ψ t L 2 (I,W ) C ( i σ,t σ t L 2 (I,W ) + I J L(W ) σ t L 2 (I,W )). (4.18) We may furter combine tese terms, wic we sall do in a separate corollary, but it is useful to keep tings separate, wic allows terms to be analyzed individually wen considering specific coices of V and V. Te error terms i σ σ and i ũ u and teir time derivatives are furtermore estimated in terms of best approximation norms and variational crimes via te teory of Holst and Stern [22]. Te different Bocner norms involved arise from te structure of te error evolution equations. Proof. We adapt te proof tecnique in [33, 20] to our situation, and for ease of notation, unsubscripted norms will denote te W -norms and norms subscripted wit just will denote norms on te approximating complex. We now assemble te estimates above separately by computing te W -norms of te errors and teir differentials. We begin by estimating θ(t). We use te standard tecnique of using te solutions as teir own test functions: Set ϕ = θ and ω = ε in (4.12). Adding te two equations togeter yields 1 d 2 dt θ 2 + ε 2 + dθ 2 = ρ t + p + (Π i )u t, θ, t I (4.19) Following Tomée [33], we introduce δ > 0 to account for non-differentiability at θ = 0, and observe tat ( θ 2 + δ 2 ) 1/2 d dt ( θ 2 + δ 2 ) 1/2 = 1 d 2 dt ( θ 2 + δ 2 ) = 1 d 2 dt θ 2 ( ρ t + p + (Π i )u t ) θ, using (4.19), te Caucy-Scwarz inequality, and te definition of operator norms (our goal is to get all of tose quantities on te rigt side of te equation close to zero, so we need not care too muc about teir sign). Tus, since θ ( θ 2 + δ2 ) 1/2, we ave, canceling θ, d dt ( θ 2 + δ 2 ) 1/2 ρ t + p + (Π i )u t. Now, using te Fundamental Teorem of Calculus, we integrate from 0 to t to get t d θ(t) = θ(0) + lim δ 0 0 dt ( θ 2 + δ 2 ) 1/2 ( ρ t + p + (Π i )u t ). 0 (4.20) θ(0) vanises by our coice of initial condition as te elliptic projection. Next, continuing to follow [20], we consider ε(t). We differentiate te first error equation and substitute ϕ = 2θ t and ω = 2ε, so tat t ε t, 2ε θ t, 2dε = 0 (4.21) θ t, 2θ t + dε, 2θ t + dθ, 2dθ t = ρ t + p + (Π i )u t, 2θ t. (4.22)

25 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 25 Adding te two equations as before, we ave, by Caucy-Scwarz and te AM-GM inequality, d dt ε θ t 2 + d dt dθ 2 2 ρ t θ t + 2 p θ t + 2 (Π i )u t θ t 2 ( ρ t 2 + p 2 + (Π i )u t ) θ 2 t 2. Again, dropping some positive terms (tis time θ t 2 ), using te Fundamental Teorem of Calculus and noting te initial conditions vanis by te coice of elliptic projection, we ave t ( ) ε 2 + dθ 2 2 ρt 2 + p 2 + (Π i )u t 2. (4.23) 0 Finally, we estimate dε. As in te estimate above, we differentiate te first equation wit respect to time, and substitute ω = 2ε t, ϕ = 2dε t, ε t, 2ε t θ t, 2dε t = 0 (4.24) θ t, 2dε t + dε, 2dε t + dθ, 2ddε t = ρ t + p + (Π i )u t, 2dε t. (4.25) Noting tat d 2 = 0, p is perpendicular to te coboundaries, and ψ = d ρ, we add te equations to get 2 ε 2 + d dt dε 2 = 2 ρ t + (Π i )u t, dε t = 2 ψ t + d (Π i )u t, ε t ψ t 2 + d (Π i )u t ε 2. By te Fundamental Teorem of Calculus, and noting vanising initial conditions (and an exact cancellation of positive terms), we ave dε 2 t 0 ( ) ψt 2 + d (Π i )u t 2. (4.26) We now estimate ρ and ψ. We note tat te time derivative of te solutions are also solutions to te mixed formulation, at least provided tat u t and oter associated quantities are sufficiently regular (in te domain of te Laplace operator) for te norms and derivatives to make sense. Ten (recalling i + = J 1 i ), we ave and ρ(t) = ũ i u ũ i + u + i+ u i u ψ(t) = σ i σ σ i + σ + i+ σ i σ i + ( i ũ u + I J u ), (4.27) i + ( i σ σ + I J σ ). (4.28) Te same estimates old for te time derivatives. Te first terms are te estimates tat allow us to use te teory of 2.2. We note tat te teory acutally uses V -norms, but it will work. We cannot improve tis in te abstract teory; instead, we use teory for specific coices of V, W, and V, suc as appropriately cosen de Ram complexes and approximations to improve te estimates ([3, 3.5], [1, Teorem 3.1]). For tese cases, it is elpful to keep te individual estimates on ε 2, θ 2, etc. separated. We ave combined terms because te abstract teory gives us all te variational crimes togeter, as it makes eavy use of te bilinear forms above. Additional improvement of estimates

26 26 M. HOLST AND C. TIEE based on regularity as done in [3] cannot made for te variational crimes, as discussed in [22, 3.4]. We give te relevant example and result in te next section. Corollary 4.3 (Combined L 1 estimate). Let θ, ρ, ψ, and ε be as above. Ten we ave i σ σ L 1 (I,V ) + i u u L 1 (I,V ) C ( ρ t L 2 (I,W ) + (Π i )u t L 2 (I,W ) + ψ t L 2 (I,W ) + d (Π i )u t L 2 (I,W ) + p L 2 (I,W ) + i σ σ L 2 (I,V ) + i ũ u L 2 (I,V )). (4.29) Furter expanding te time derivative terms, we ave i σ σ L 1 (I,V ) + i u u L 1 (I,V ) C ( i ũ,t u t L 2 (I,W ) + i σ,t σ t L 2 (I,W ) + I J u t L 2 (I,W ) + I J σ t L 2 (I,W ) + (Π i )u t L 2 (I,W ) + d (Π i )u t L 2 (I,W ) + i p L 2 (I,W ) + i σ σ L 2 (I,V ) + i ũ u L 2 (I,V )). Tese terms are organized as follows: te W -error in te approximations of te time derivatives, te variational crimes wit I J, te data approximation error for te time derivatives, and finally te V -approximation errors for te elliptic projection. Tese can be furter expanded in terms of best approximation errors, but we will not ave use for tat outside of specific examples were te computation is easier done wit te previous teorems. Tis corollary is simply stated for conceptual clarity and a qualitative sense of all te different individual contributions to te error. Proof. First, we note tat by te Caucy-Scwarz inequality, te estimate for dθ (4.14) can be rewritten as using L 2 (I, W ) norms to matc te squared terms (4.23) and (4.26). Combining and absorbing constants, we arrive at i σ (t) σ(t) V + i u (t) u(t) V C ( ρ t L 2 (I,W ) + (Π i )u t L 2 (I,W ) ) + ψ t L 2 (I,W ) + d (Π i )u t L 2 (I,W ) + p L 2 (I,W ) + i σ (t) σ(t) V + i ũ (t) u(t) V. Integrating from 0 to T, te latter two V -norm terms become L 1 (I, V ) norms (and absorb te factor of T from integrating te first into te constant). Finally, using Caucy- Scwarz to cange te L 1 (I, V ) norm into an L 2 (I, V ) norm, and substituting for ρ t and ψ t gives te result. 5. PARABOLIC EQUATIONS ON COMPACT MANIFOLDS As an application of te preceding results, we return to our original motivating example of de Ram complex to explore an example wit te Hodge eat equation on ypersurfaces of Euclidean space, generalizing te discussion in [22, 20]. Let M be compact ypersurface embedded in R n+1. M inerits a Riemannian metric from te Euclidean metric of R n Te de Ram Complex on a Manifold. We define te L 2 differential k-forms on M given by { } L 2 Ω k (M) := a i1...i k dx i 1 dx i k Ω k (M) : a i1...i k L 2 (M), 1 i 1 < <i k n

27 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 27 te standard indexing of differential form basis elements, namely strictly increasing sequences from {1,..., n}. Te inner product is given by ω, η = ω η, were is te Hodge operator corresponding to te metric. Te weak exterior derivative d k is defined on te domains HΩ k (M), and we ave a Hilbert complex (L 2 Ω, d) wit domain complex (HΩ(M), d), wit d k+1 d k = 0: 0 HΩ 0 d 0 HΩ 1 d 1 dn 1 HΩ n 0. As required in te abstract Hilbert complex teory, eac domain space carries te grap inner product: u, v H Ω k (M) := u, v L 2 Ω k (M) + d k u, d k v L 2 Ω k+1 (M). For open subsets U R n, te ends (k = 0 and k = n) of tis complex are familiar Sobolev spaces of vector fields wit te traditional gradient, curl, and divergence operators of vector analysis: 0 H 1 (U) grad H(curl) curl H(div) div L 2 (U) 0. Similarly, te dual complex is H Ω(M) defined by H Ω k (M) := HΩ n k (M), consisting of Hodge duals of (n k)-forms. We ave tat te embedding HΩ k (M) H Ω k (M) L 2 Ω k (M) is compact, wic enables a Poincaré Inequality to old and te resulting Hilbert complex (L 2 Ω k (M), d) to be a closed complex [29, 3]. To summarize, we ave te following: Teorem 5.1. Let M be a compact Riemannian ypersurface in R n+1. Ten taking W k = L 2 Ω k (M), wit maps d k te exterior derivative defined on te domains V k = HΩ k (M), (W, d) is a closed Hilbert complex wit domain (V, d). We tus are able to define Hodge Laplacians, and see all te abstract teory for te continuous problems (2.15) and (3.6) applies wit tese coices of spaces Approximation of a ypersurface in a tubular neigborood. In order to approximate te problems (2.15) and (3.6), we consider, following [22], a family of approximating ypersurfaces M to an oriented ypersurface M all contained in a tubular neigborood U of M. Te surfaces M generally will be piecewise polynomial (say, of degree s); te case s = 1 corresponds to (piecewise linear) triangulations, studied in [16, 14], and generalized for s > 1 in [13]. However, te piecewise linear case still is instrumental in te analysis and indeed, te definition of te spaces (via Lagrange interpolation), and so we sall denote it by T (te triangulation, i.e., set of simplices, will be correspondingly denoted by T, and teir images under te interpolation will be denoted ˆT ). It is convenient, also, to assume tat te vertices of te bot te triangulation and te iger-degree interpolated surfaces actually lie on te true ypersurface. Te normal vector ν to te M allows us to define a signed distance function dist : U R given by dist(x) = ± dist(x, M) = ± inf x y y M were te sign is cosen in accordance to wic side of te normal x lies on. By elementary teorems in Riemannian geometry [15, C. 6], dist is smoot, provided U is small enoug; te maximum distance for wic it exists is controlled by te sectional curvature of M. Te normal ν can be extended to te wole neigborood; in fact it is te gradient δ. It is also convenient to define te normals ν to te approximating surfaces M. In most of te examples we consider, we assume te vertices of M (and T )

28 28 M. HOLST AND C. TIEE U a(x) δ(x) ν x M M FIGURE 1. A curve M wit a triangulation (blue polygonal curve M ) witin a tubular neigborood U of M. Some normal vectors ν are drawn, in red; te distance function δ is measured along tis normal. Te intersection x of te normal wit M defines a mapping a from x to its base point a(x) M. lie on M, but tis is not a strict requirement. Instead, we need a condition to ensure tat te ypersurfaces M are diffeomorpic to M, eliminating te possibility of a double covering (e.g., as pictured in [17, Fig. 1, p. 12]). In particular, we want M to ave te same topology as M. Tis is again restriction on te size of te tubular neigborood. In suc a neigborood U, every x U decomposes uniquely as x = a(x) + dist(x)ν(x), (5.1) were a(x) M, and a : U M is in fact a smoot function, called te normal projection. a can ten be used to define te degree-s Lagrange interpolated ypersurfaces by considering te image of T under te degree-s Lagrange interpolation of a over eac simplex in T (we write a k : T M for tis) [13, 2.3]. Now, Holst and Stern [22] sow, for ypersurfaces, te following result for te variational crime I J : Teorem 5.2 (Holst and Stern [22], Teorem 4.4). Let M be an oriented, compact m-dimensional ypersurface in R m+1, and M be a family of ypersurfaces lying in a tubular neigborood U of M transverse to its fibers, suc tat δ 0 and ν ν 0 as 0. Ten for sufficiently small, I J C( δ + ν ν 2 ). (5.2) A result of Demlow [13, Proposition 2.3] states tat, in te case tat M is obtained by degree-s Lagrange interpolation, tat δ < C s+1 and ν ν < C s. Tus, putting tese results togeter, we ave tat I J C s+1. (5.3) Now, te tree best approximation error terms (2.17) for finite element approximation by polynomials of degree r are bounded by C r, C r+1, or C r 1, depending on te component cosen, so it is crucial to allow for tis case, and te convergence rate is optimal wen r = sfigure 2 also dramatically demonstrates ow muc better a igerorder approximation can be wit a given mes size.

29 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS FIGURE 2. Approximation of a quarter unit circle (black) wit a segment (blue) and muc better quadratic Lagrange interpolation for te normal projection (red), toug te underlying triangulation is te same (and tus also te mes size). Restricting a to te surfaces M gives diffeomorpisms a M : M M. a : M M is terefore a diffeomorpism wen restricted to eac polyedron (and is at least globally Lipscitz continuous, te maximum degree of regularity in te piecewise linear case. Tis is not a problem for Hodge teory, because te form spaces are at most H 1 were regularity is concerned; see [36]). See Figure Finite element spaces. We tus coose finite-dimensional subspaces Λ k of HΩk (M ) for eac k, satisfying te subcomplex property d Λ k Λk+1. We can ten pull forms on M back to forms on M via te inverse of te normal projection, wic furnises te injective morpisms i k : Λk HΩk (M) (since pullbacks commute wit d) required by te teory above in Section 2. Te main finite element spaces relevant for our purposes are two families of piecewise polynomials, discussed in detail in [2, 3]. We must coose tese spaces for our equations in a specific relationsip in order for te numerical metods and teory detailed above to apply, and for te approximations to work. Tis is wy we prefer a piecewise polynomial approximation of M as opposed to a curved triangulation of M itself; tese are sown to ave tese necessary properties. Definition 5.3 (Polynomial differential forms). Let P r denote polynomials of degree at most r, in n variables, and H r be te subspace of omogeneous polynomials. We define te first family, denoted P r Λ k (T ), to consist of all k-forms wit coefficients belonging to P r wen restricted to eac n-simplex of T. Te continuity condition is tat te polynomials on two simplices aving a common face must ave te same trace to tat face. Te second family, denoted P r Λ k (T ), are intermediate spaces, between te spaces of te first class: P r 1 Λ k (T ) P r Λ k (T ) P r Λ k (T ).

30 30 M. HOLST AND C. TIEE Tese are defined as follows: first, consider te radial vector field X = x i x i, tat is, at eac x, it is a radially pointing vector of lengt x, and ten define te Koszul operator κω := X ω, te interior product wit X. Ten P r Λ k (T ) := P r 1 Λ k κh r 1 Λ k+1. Tis is a direct sum, since κ always raises polynomial degree and decreases form degree, so yields omogeneous polynomials of degree r. κ is in some ways dual to te operator d (wic, in particular, increases form degree and decreases polynomial degree), and by te properties of interior products, κ 2 = 0. Tese polynomial spaces generalize existing finite element spaces, suc as Witney forms, Nédélec elements, and Raviart-Tomas elements (see [20, 3] for tese examples and more), realizing te collection and clarification of previous results respecting vector metods, as we ave mentioned numerous times trougout tis work. Te important property of tese spaces is tat tey admit te cocain projections wose role we ave seen is so important in te teory. First, we describe te case were M = U is a domain in R n wit smoot or Lipscitz boundary. π k : L 2 Ω k Λ k were Λ k {P r Λ k (T ), P r Λ k (T )}. (5.4) Tese operators, by virtue of teir construction, are uniformly bounded (in L 2 Ω k, not just HΩ k ) wit respect to. Finally, te following teorem explicitly expresses te projection error (and ence, best approximation error) in terms of powers of te mes size and te norms of te solution. Teorem 5.4 (Arnold, Falk, and Winter [3], Teorem 5.9). (i.) Let Λ k be one of te spaces P r+1λ k (T ) or, if r 1, P r Λ k (T ). Ten π k is a cocain projection onto Λ k and satisfies ω π k ω L 2 Ω k (U) c s ω H s Ω k (U), ω H s Ω k (U), for 0 s r + 1. Moreover, for all ω L 2 Ω k (U), π k ω ω in L2 as 0. (ii.) Let Λ k be one of te spaces P rλ k (T ) or P r Λ k (T ) wit r 1. Ten d(ω π k ω) L 2 Ω k (U) c s dω H s Ω k (U), for 0 s r. ω H s Ω k (U), Tese bounded cocain operators are explicitly constructed in [2, 3]; tey are te natural interpolation operators defined for continuous differential forms and analogous to polynomial interpolation operators on functions, but combined wit smootings to allow extension to H s differential forms wic may not necessarily be continuous. Example 5.5 (Te Mixed Hodge Laplacian problem on an open subset of R n ). For te mixed Hodge Laplacian problem we considered above, we must coose Λ k 1 and Λ k in suc a manner suc tat dλ k 1 Λ k ; one cannot make te coices of spaces completely independent of one anoter for our mixed problem [3, 5.2]. For example, if we coose = P r Λ k 1 (T ), we necessarily must coose Λ k 1 Similarly, for Λ k 1 Λ k { P r 1 Λ k (T ), P r Λ k (T ) }. = P r Λ k 1 (T ), we coose Λ k { P r Λ k (T ), P r 1 Λ k (T ) }. Continuing in tis manner down te complex, tere are 2 n possible full cocain subcomplexes one can form wit tese coices of spaces. Of course, for one single Hodge

31 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 31 Laplacian problem, we only need to work wit tree spaces in te cain, since te equations only involve (k 1)- and k-forms and teir differentials. Example 5.6 (Finite Element Spaces on Riemannian manifolds). Now, suppose we are back in te situation wit a Riemannian ypersurface M R n+1, wit a family of degree-s Lagrange-interpolated surfaces M, over a triangulation T. We can still consider te polynomial finite element spaces on te triangulation T as before; te only difference ere is tat te simplices may not join up smootly (i.e., as a manifold, it may ave corners). Tis is not a problem, because te continuity conditions enforced by te finite element spaces also allow for discontinuities or non-classical-differentiability on te simplicial boundary faces. To define te analogous polynomial spaces on te possibly curved triangulations M, we simply say a form is in te analogous polynomial spaces P r Λ k ( ˆT ) if its pullback by te inverse of te interpolated normal projections a k : T M to T is in P r Λ k (T ) [13, 2.5]. Now, from P r Λ k ( ˆT ), we pull tese forms back to te surface M via te normal projections (a M ) 1. Tis gives te injective morpisms i k : Λk HΩk (M); it commutes wit te differentials, since te pullbacks do. For te bounded cocain operators, te situation is similar. We ave π k : HΩk (M ) Λ k a cocain projection defined by pulling forms defined in neigboroods back to te triangulations (using te trace teorem if necessary), as constructed in [2, 3]. Ten we compose wit te pullbacks (a M ). Tis gives us te cocain projections π k : HΩ k (M) Λ k (by [22, Teorem 3.7]) Estimates for te Mixed Hodge Laplacian problem on manifolds. Wit tis, we can ten integrate te terms from [22, Example 4.6] to get te results for te parabolic equations (or, equivalently, add te variational crimes to [20, 1]). Let us consider now te mixed Hodge Laplacian problem on Riemannian ypersurfaces, considering te setup in te previous example. Namely, we consider W k = L 2 Ω k (M), V k = HΩ k (M) as above, te approximating spaces V k 1 = P r+1 Λ k 1 ( ˆT ) and V k = P rλ k ( ˆT ), and finally te inclusion and projection morpisms as above (possibly wit additional pullbacks for interpolation degree s > 1). Of course, as mentioned before, tese are not te only ways of coosing te spaces, but we stay wit, and make estimates based on, tis coice for te remainder of tis example (te same coice made in [22, Example 4.6]). For a function f L 2 Ω k (M), we ave an approximate solution (σ, u, p ) i X to te elliptic problem, on te true subcomplex i W (wit modified inner product, as in te teory of 2.4). For f sufficiently regular, and (σ, u, p) satisfying te regularity estimate [3, 20] u H s+2 + p H s+2 + σ H s+1 C f H s, (5.5) for 0 s s max, ten, since we are in te de Ram complex, were te cocain projections are W -bounded, we ave te improved error estimates of Arnold, Falk, and Winter [3, 3.5 and p. 342] for te elliptic problem: u i u + p i p C r+1 f H r 1 (5.6) d(u i u ) + σ i σ C r f H r 1 (5.7) d(σ i σ ) C r 1 f H r 1. (5.8) We sould also note tat Arnold and Cen [1] prove tat tis also works for a nonzero armonic part [1, Teorem 3.1]. Holst and Stern [22] augment tese estimates to include te variational crimes, so tat (canging te notation to suit our problem) for

32 32 M. HOLST AND C. TIEE ( σ, ũ, p ) X, te discrete solution to te elliptic problem now on te approximating complexes we ave cosen, we ave te estimates u i ũ + p i p + ( d(u i ũ ) + σ i σ ) + 2 d(σ i σ ) C( r+1 f H r 1 + s+1 f ). (5.9) We note te terms associated to te different powers of above correspond exactly to te breakdown (4.14)-(4.16) above. For te elliptic projection in our problem, we also need to account for te nonzero armonic part of te solution. Setting w = P H ũ and w = Π w, we ave tat our tree additional terms (given by Teorem 2.16 above) are te corresponding best approximation error inf v V k w v V, te I J term, and te data approximation w i w. For te best approximation, we make use of our observation about te inequality (2.24), in wic we may instead use te W -norm instead of te V -norm in te case tat te projections are W -bounded, as tey are ere in te de Ram complex. Because w is armonic, it is smoot (and in particular, in H r+1 ), so we may apply Teorem 5.4 to find tat it is of order C r+1 w H r+1. Te I J term as already been sown to be of order C s+1 above in Teorem 5.2. Finally, by Teorem 2.18 above, we ave tat data approximation splits into te oter two terms. Terefore, to summarize, we ave Teorem 5.7 (Estimates for te elliptic projection). Consider (σ(t), u(t)), te solution to te parabolic problem (3.6) and (σ (t), u (t)) te semidiscrete solution in (4.1) above. Ten we ave te following estimates for te elliptic projection ( σ, ũ, p ): u i ũ + i p + ( d(u i ũ ) + σ i σ ) + 2 d(σ i σ ) C ( r+1 ( u H r 1 + w H r+1) + s+1 ( u + w ) ). (5.10) (We note p = P H ( u) = 0.) We now would like use te our main parabolic estimates to analyze te analogous quantity u(t) i u (t) + ( d(u(t) i u (t)) + σ(t) i σ (t) )+ 2 d(σ(t) i σ (t)), (5.11) and its integral, i.e. Bocner L 1 norm. Teorem 5.8 (Main combined error estimates for Riemannian ypersurfaces). Let (σ(t), u(t)), (σ (t), u (t)), and all terms involving te elliptic projection are defined as above, and te regularity estimate (5.5) is satisfied. Ten u i u L 1 (W ) + ( d(u i u ) L 1 (W ) + σ i σ L 1 (W )) + 2 d(σ i σ ) L 1 (W ) C [ ( r+1 (T + 1) ( ( )) u L 1 (H r 1 ) + w L 1 (H )) r+1 + T ut L 1 (H r 1 ) + w t L 1 (H r+1 ) + s+1 ( (T + 1) ( u L 1 (W ) + w L 1 (W )) + T ( ut L 1 (W ) + w t L 1 (W )))]. (We abbreviate L p (I, X) as L p (X).) Te constants T, of course, can be furter rolled into te constant C. We remark tat in previous results, factors of T sow up on te u t terms, and, euristically speaking, tis is due to te u t being a pysically different quantity, namely, a rate of cange. However, te appearance of te factor of T on te u comes from te armonic approximation error p, wic is, pysically speaking, a armonic source term. Te details depend on te nature of te approximation operators Π.

33 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 33 Proof. By te triangle inequality, we ave tat (5.11) breaks up into someting of te form (5.9) (taking ( σ, ũ, p ) to be elliptic projection wit f = u(t) and p = 0; ere f is not to be confused wit te parabolic source term f(t)) and i ( θ(t) + ( ε(t) + du(t) ) + 2 dε(t) ), (5.12) recalling te error quantities defined in (4.8)-(4.11). Now, substituting our estimates (4.14)-(4.16), we ten ave θ(t) ρ t L 1 (W ) + p L 1 (W ) + (Π i )u t L 1 (W ) C ( ) i ũ,t u t L 1 (W ) + p L 1 (W ) + I J u t L 1 (W ) + (Π i )u t L 1 (W ) C 1 ( ) r+1 u L 1 (H r 1 ) + u t L 1 (H r 1 ) + w L 1 (H r+1 ) + w t L 1 (H r+1 ) + C 2 s+1 ( u L 1 (W ) + u t L 1 (W ) + w L 1 (W ) + w t L 1 (W )). (5.13) For dθ + ε, te computation is almost exactly te same, except wit possibly different constants, to account for using L 2 Bocner norms, and tat : dθ(t) + ε(t) C ( ) i ũ,t u t L 2 (W ) + p L 2 (W ) + I J u t L 2 (W ) + (Π i )u t L 2 (W ) C 3 ( ) r+1 u L 2 (H r 1 ) + u t L 2 (H r 1 ) + w L 2 (H r+1 ) + w t L 2 (H r+1 ) + C 4 s+1 ( u L 2 (W ) + u t L 2 (W ) + w L 2 (W ) + w t L 2 (W )). Tese terms are actually absorbed into te lower order terms by te extra factor of, due to consisting entirely of te same order terms except using a different norm. However, te situation is sligtly different for dε ; namely we use (5.7) to get a term of order r, and te d on te variational crime part also removing a factor of : dε(t) C ( ) ψ t L 2 (W ) + d (Π i )u t L 2 (W ) C ( ) i σ,t σ t L 2 (W ) + I J σ t L 2 (W ) + d (Π i )u t L 2 (W ) C 5 r ( u t L 2 (H r 1 ) + w L 2 (H r+1 )) + C 6 s ( u L 2 (W ) + u t L 2 (W ) + w L 2 (W ) + w t L 2 (W )) However, we see tat multiplying by 2, tis term also gets absorbed; tus we need only consider te error from dθ in furter calculation of te combined estimate. We ave, tus far: u(t) i u (t) + ( d(u(t) i u (t)) + σ(t) i σ (t) ) + 2 d(σ(t) i σ (t)) C 1 ( ) r+1 u L 1 (H r 1 ) + u t L 1 (H r 1 ) + w L 1 (H r+1 ) + w t L 1 (H r+1 ) + C 2 s+1 ( u L 1 (W ) + u t L 1 (W ) + w L 1 (W ) + w t L 1 (W )) + C ( r+1 ( u(t) H r 1 + w(t) H r+1) + s+1 ( u(t) + w(t) ) ). (5.14) Integrating wit respect to t from 0 to T, we find tat te already-present Bocner norms are constant and tus introduce an extra factor of T. Absorbing te constants except T gives te result. Tis sows, in particular, tat te optimal rate of convergence occurs wen r = s, i.e., te polynomial degree of te finite element functions matces te degree of polynomials used to approximate te ypersurface. Tis tells us, for example, it is not beneficial to use iger-order finite elements on, say, a piecewise linear triangulation. Finally, to put

34 34 M. HOLST AND C. TIEE tese estimates into some perspective and elp develop some intuition for teir meaning, we present te generalization of te estimates of Tomée from te introduction. Corollary 5.9 (Generalization of [33, 20, 1]). Focusing on just te components u and σ separately, we ave te following estimates (assuming te regularity estimates (5.5) are satisfied), and supposing r = s, i.e., te finite element spaces considered consist of polynomials of te same degree as te interpolation on te surface: t ) u(t) i u (t) C ( u(t) r+1 H r+1 + ( u(s) H r+1 + u t (s) H r+1) ds ( t σ(t) i σ (t) C ( u(t) r+1 H r ( u(s) 2 H r+1 + u t(s) 2 H r+1 ) ds ) 1/2 ) Tis easily leads to an estimate in a Bocner L norm (simply take te sup in te non-bocner norm terms and t = T in te integrals); tis sows tat te error in time is small at every t I. Similar estimates old for L 2 (I, W ) norms. Proof. We consider te improved error estimate and variational crimes in u and σ separately. We first ave, by expanding te terms in (4.20) as in te derivation of (5.13), u(t) i u (t) C ( u(t) i ũ (t) + θ(t) ) + t 0 C ( u(t) r+1 H r 1 + w(t) H r+1 ) ( u(s) H r 1 + u t (s) H r 1 + w(s) H r 1 + w t (s) H r 1) ds. Te result follows by noting tat u H r+1 includes estimates on all te second derivative terms in u, and w = P H u, so tose two norms can all be combined (wit possibly different constants). Next, we consider σ. Te improved error estimates [3, p. 342] imply tat if we do not combine estimates involving du wit tose of σ for te modified solution, and f is regular enoug to use te H r - rater tan H r 1 -norm, ten we can gain back one factor of, so tat it is of order r+1 (rater tan r as in (5.7)). On te oter and, te elliptic projection error ε(t) still can be taken along wit dσ(t) and was of order r+1 to begin wit. Tus, applying (4.23), we ave σ(t) i σ (t) C ( σ(t) i σ (t) + ε(t) + du(t) ) [ t + 0 C r+1 ( u(t) H r + w(t) H r+1 ( u(s) 2 H r 1 + u t(s) 2 H r 1 + w(s) 2 H r 1 + w t(s) 2 H r 1 ) ds ] 1/2 ) ( t C ( u(t) r+1 H r ( u(s) 2 H r+1 + u t(s) 2 H r+1 ) ds ) 1/2 ), were we ave used te same consolidation tecniques for te norms on u and w into norms on u as before. We see te variational crimes (arising from te extra p ) account for te sole additional term in te integrals. Tis cannot be improved witout furter information on te projections Π. Oterwise, for r = 1, wic correspond to piecewise linear discontinuous elements for 2-forms (u), and piecewise quadratic elements for 1-forms (σ) wit normal

35 FEEC FOR EVOLUTION PROBLEMS ON MANIFOLDS 35 continuity (Raviart-Tomas elements), as studied by Tomée, we obtain te estimates e derived (and since te p is not tere in is case, we ave tat te extra terms wit u do not appear under te integral sign). 6. A NUMERICAL EXAMPLE In order to actually simulate a solution to te Hodge eat equation, we consider te scalar eat equation on a domain in M R 2, but now using a mixed metod wit 2- forms rater tan te functions. We return to te evolution equation for bot σ and u, (3.6) above, wic we recall ere: σ t, ω + dσ, dω = f, dω, ω V k 1, t I, u t, ϕ + dσ, ϕ + du, dϕ = f, ϕ, ϕ V k, t I, u(0) = g. (6.1) Given S V k = HΩ 2 (M) and H V k 1 = HΩ 1 (M), we coose bases, and use te semidiscrete equations (1.4), wic we recall ere (setting U to be te coefficients of u in te basis for S, and Σ to be te coefficients of σ in te basis for H ) d dt ( ) ( D B T Σ = 0 A U) ( 0 0 B K) ( Σ U ) ( ) 0 + F (6.2) Tis may be discretized via standard metods for ODEs. For our implementation, we use te backward Euler metod. Tis means we consider sequences (Σ n, U n ) in time, and ten rewrite te derivative instead as a finite difference, ( evaluating ) te vector field D B T portion on te rigt side at timestep n + 1, taking M = : 0 A (( ) ( )) ( ) ( ) 1 Σ t M n+1 Σ n 0 0 Σ n+1 U n+1 U n = B K U n+1 + or ( ( )) ( ) 0 0 Σ n+1 M + t B K U n+1 = M ( 0 F n+1 ) ( ) ( ) Σ n 0 U n + t F n+1. We now ave written te system as a sparse matrix times te unknown (Σ n+1, U n+1 ). Tis allows us to solve te system directly using sparse matrix algoritms witout explicitly inverting any matrices, making te iterations efficient. To analyze te error of te approximations, we can combine te above error estimates wit te standard error analysis of Euler metods. See Figure CONCLUSION AND FUTURE DIRECTIONS We ave seen tat te abstract teory of Hilbert complexes, as detailed by Arnold, Falk, and Winter [3], and Bocner spaces, as detailed in Gillette and Holst [20] and Arnold and Cen [1], as been very useful in clarifying te important aspects of elliptic and parabolic equations. Te mixed formulation gives great insigt into questions of existence, uniqueness, and stability of te numerical metods (linked by te cocain projections π ). Te metod of Tomée [33] allows us to leverage te existing teory for elliptic problems to apply to parabolic problems, taking care of te remaining error terms by te use of differential inequalities and Grönwall estimates (in te important error evolution equations (4.12) above). Incorporating te analysis of variational crimes allow us to carry tis teory over to te case of surfaces and teir approximations. We remark on some possible future directions for tis work. Some existing surface finite elements for parabolic equations ave been studied by Dziuk and Elliott [17], and

36 36 M. HOLST AND C. TIEE ( A ) 1 second ( B ) 2 seconds ( C ) 3 seconds ( D ) 4 seconds F IGURE 3. Hodge eat equation for k = 2 in a square modeled as a mes, using te mixed finite element metod given above. Initial data is given as te (discontinuous) caracteristic function of a C-saped set in te square. Te timestepping metod is given by te backward Euler discretization, wit timestep t = muc oter work by Dziuk, Elliott, Deckelnick [12, 11], wic actually treat te case of an evolving surface, and treat a nonlinear equation, te mean curvature flow. Generally speaking, tis translates to an additional time dependence for evolving metric coefficients, and a logical place to start is in te Tome e error evolution equations (4.12). Nonlinear evolution equations for evolving metrics also suggests te Ricci flow [27, 9, 10], instrumental in sowing te Poincare conjecture. Te callenge tere, besides nonlinearity, is tat tensor equations do not necessarily fit in te framework for FEEC. On te oter and, te Yamabe flow [31], wic solves for a conformal factor for te metric (and is equivalent to te Ricci flow in dimension 2) suggests an interesting nonlinear scalar evolution equation for wic tis analysis may be useful. Gillette, Holst, and Zu [20] also analyzed yperbolic equations in tis framework, and it would be interesting and useful to analyze metods on surfaces (including te evolving case), as well as taking a more integrated approac in spacetime. Tis is usually taken care of using te discrete exterior calculus (DEC), te finite-difference counterpart to FEEC to analyze yperbolic equations [26].