Institut für Numerisce und Angewandte Matematik Hybrid discontinuous Galerkin metods wit relaxed H(div-conformity for incompressible flows. Part I Lederer, P. L., Lerenfeld, C., Scöberl, J. Nr. 4 Preprint-Serie des Instituts für Numerisce und Angewandte Matematik Lotzestr. 16-18 D - 3783 Göttingen
HYBRID DISCONINUOUS GALERKIN MEHODS WIH RELAXED H(DIV-CONORMIY OR INCOMPRESSIBLE LOWS. PAR I PHILIP L. LEDERER, CHRISOPH LEHRENELD, AND JOACHIM SCHÖBERL Abstract. We propose a new discretization metod for te Stokes equations. e metod is an improved version of te metod recently presented in [C. Lerenfeld, J. Scöberl, Comp. Met. Appl. Mec. Eng., 361 (216] wic is based on an H(div-conforming finite element space and a Hybrid Discontinuous Galerkin (HDG formulation of te viscous forces. H(div-conformity results in favourable properties suc as pointwise divergence free solutions and pressure-robustness. However, for te approximation of te velocity wit a polynomial degree k it requires unknowns of degree k on every facet of te mes. In view of te superconvergence property of oter HDG metods, were only unknowns of polynomial degree k 1 on te facets are required to obtain an accurate polynomial approximation of order k (possibly after a local post-processing tis is sub-optimal. e key idea in tis paper is to sligtly relax te H(div-conformity so tat only unknowns of polynomial degree k 1 are involved for normal-continuity. is allows for optimality of te metod also in te sense of superconvergent HDG metods. In order not to loose te benefits of H(div-conformity we introduce a ceap reconstruction operator wic restores pressure-robustness and pointwise divergence free solutions and suits well to te finite element space wit relaxed H(div-conformity. We present tis new metod, carry out a toroug -version error analysis and demonstrate te performance of te metod on numerical examples. Key words. Stokes equations, Hybrid Discontinuous Galerkin metods, H(div-conforming finite elements, pressure robustness AMS subject classifications. 35Q3, 65N12, 65N22, 65N3 1. Introduction and structure of te paper. In te recent paper [25] a new finite element discretization metod for te incompressible Navier-Stokes equations as been presented. It is based on an efficient time integration sceme wic allows to split te Navier-Stokes problem into two simpler subproblems, a Stokes-type problem and a yperbolic transport problem. or bot subproblems specifically taylored metods ave been applied. In tis work we turn our attention to te numerical treatment of te Stokes problem in a velocity-pressure formulation: { ν u + p = f in Ω, (1.1 div(u = in Ω, wit boundary conditions u = on Γ D Ω. Here, ν = const is te kinematic viscosity, u te velocity, p te pressure, and f is an external body force. At te eart of te present contribution lies te modification of te previously considered H(div-conforming ybrid discontinuous Galerkin finite element metod resulting in a reduction of te gloablly coupled unknowns occuring in te linear systems. Beside oter purposes, te aim of abandoning H 1 -conformity and consider an H(div- conforming discontinuous Galerkin (DG metod instead is to find a suitable approximation of te incompressibility constraint. is leads to a discontinuity only for te tangential part of te velocity and was introduced in [1, 11] and for te Brinkman Problem in [2]. Neverteless DG metods are considered to be muc more expensive compared to a continuos Galerkin (CG metod wen it comes to solving linear Institute for Analysis and Scientific Computing, U Wien, Wien, Austria; email: {pilip.lederer,joacim.scoeberl}@tuwien.ac.at Institute for Numerical and Applied Matematics, University of Göttingen, Göttingen, Germany; email: lerenfeld@mat.uni-goettingen.de 1
systems. is is mainly due to te dramatically increased numbers of globally coupling degrees of freedom. An approac to overcome tis drawback is te concept of ybridization. ere, even more unknowns are introduced on te skeleton but neverteless tis leads to two major benefits. irst, te coupling between elements is reduced and secondly, te structure of te coupling allows for static condensation of te element unknowns, see [8, 25, 24]. Hybrid discontinuous Galerkin (HDG metods are also of interest due to teir capability of providing a superconvergent post processing. is step allows to reconstruct interior element unknowns resulting in an accuracy of order k + 1 in te volume wen we use polynomials of order k on te skeleton. In [25] a similar result was acieved by te usage of projected jumps. e idea is to use only polynomials of order k 1 on te skeleton and introduce a projection on te occurring jump terms. is reduction of te polynomial order as no impact on te accuracy of te metod, tus can be seen as an approac to obtain superconvergency. Still, tis tecnique only results in a reduced coupling of te tangential part. is work considers an approac ow to reduce also te coupling wit respect to te normal component. Were an H(div-conforming metod demands normal continuity of te velocity, i.e. [u n] =, we only impose a relaxed continuity Π k 1 [u n] =. is allows facet jumps of te normal components in te igest order modes resulting in discrete divergence free approximations wit optimal order of convergence. A weak treatment of te incompressibility constraint using mixed finite element metods results in velocity error estimates wic dependent on te pressure, i.e. tey are not pressure robust, [17]. is may result in a bad velocity approximation wen te irrotational part of te rigt and side is dominant. Under tis perspective one also speaks of poor mass conservation, since large velocity errors are accompanied by large divergence errors. In [27] te problem is analyzed and a reconstruction is presented to retrieve pressure robustness. is reconstruction maps discrete divergence free testfunctions onto exactly divergence free test functions and is applied only on te rigt and side. is approac was performed for several elements using discontinuous pressures [27, 7, 26, 17, 28] and recently continuous pressure elements [22, 21]. or H(div-conforming finite element metods discrete incompressibility implies pointwise incompressibility and render te metods automatically pressure robust. ese metods ave been investigated for incompressible flow problems in a.o. [11, 19, 18, 25, 14, 34]. A relaxed H(div-conformity results in only discretely divergence free approximations and tus leads to a lack of pressure robustness. or tis we present a simple reconstruction and provide numerical examples revealing te benefits of te modified Stokes discretization. We note tat we only treat an -version error analysis ere. e p-version error analysis of te metod will be treated in a fortcoming paper. urtermore we want to refer to capter 1.2 in [25] for a more detailed summery of works considering a discontinuous Galerkin approac. Structure of te paper. In section 2 we introduce new finite element metods. We start wit a basic metod using an HDG formulation wit relaxed H(div-conformity. urter, we introduce a reconstruction operator to restore solenoidal velocity fields. Based on tis operator we ten define a modified pressure robust discretization. e construction of te finite element space wit relaxed H(div-conformity is nonstandard and te efficient realization of te reconstruction operator eavily relies on te coice of basis functions, we devote section 3 solely to aspects of te finite element spaces and teir bases. is section is te most tecnical part of te paper. We note 2
owever tat te remaing part of te work can be understood witout it. A toroug a priori error analysis of te proposed metods is carried out in section 4. We conclude our work wit numerical examples in section 5 wic confirm te teoretical findings. 2. Relaxed H(div-conforming HDG formulation of te Stokes problem. 2.1. Preliminaries and Notation. We begin by introducing some preliminary notation and assumptions. Let be a sape-regular triangulation of an open bounded domain Ω in R d wit a Lipscitz boundary Γ. By we denote a caracteristic mes size. We note tat can be understood as a local quantity, i.e. it can be different in different parts of te mes due to a cange in te local mes size. e local caracter of will, owever, not be reflected in te notation. e element interfaces and element boundaries coinciding wit te domain boundary are denoted as facets. e set of tose facets is denoted by and tere olds =. We separate facets at te domain boundary, exterior facets and interior facets by te sets ext, int. or sufficiently smoot quantities we denote by [ ] and { } te usual jump and averaging operators across te facet int ext. or we define [ ] and { } as te identity. We restrict to te case were te mes consists of straigt simplicial elements. urter, in te analysis we consider only te case of omogeneous Diriclet boundary conditions to simplify te presentation. By P m ( and P m ( we denote te space of polynomials up to degree m on a facet and an element, respectively. By H m (Ω we denote te usual Sobolev space on Ω, wereas H m ( denotes its broken version on te mes, H m ( := {v L 2 (Ω : v H m ( }. In te discretization we introduce element unknowns wic are supported on elements (in te volume and different unknowns wic are supported only on facets. We indicate tis relation wit a subscript for unknowns supported on facets and a subscript for unknowns tat are also supported on volume elements. or te discretization of te velocity we use bot type of functions and denote te compositions of volume and facet functions by u = (u, u. At several occasions we furter distinguis tangential and normal directions of vector-valued functions. We terefore introduce te notation wit a superscript t to denote te tangential projection on a facet, v t = v (v n n R d, were n is te normal vector to a facet. e index k wic describes te polynomial degree of te finite element approximation at many places troug out te paper is an arbitrary but fixed positive integer number. or te analysis we use te symbol x for te partial derivative wit respect to x and in a similar manner we use te sub indices (u 1 as symbol for te first component of a vectorial function u. In tis section we consider a new Hybrid DG discretization of te Stokes problem in velocity-pressure formulation. e well-posed weak formulation of te problem is: ind (u, p [H 1 (Ω] d L 2 (Ω, s.t. Ω ν u : v dx + div(vp dx = fv dx for all v [H 1 (Ω] d, Ω Ω div(uq dx = for all q L 2 (Ω. Ω (2.1 Here, we ave L 2 (Ω := {q L 2 (Ω : q dx = }. In te discretization we take Ω special care about te treatment of te incompressibility condition wic is closely 3
related to te coice of finite element spaces, wic will be introduced in te next subsection. 2.2. inite element spaces. In tis subsection we introduce te finite element spaces tat we require for our discretization. is involves several steps. irst, we introduce finite element spaces for te velocity and te pressure as tey ave been used in DG discretizations for Stokes and Navier-Stokes problems in (among oters [11, 15]. In a second step, we introduce facet unknowns for te velocity tat facilitate a more efficient treatment of arising linear systems in te spirit of Hybrid DG metods, cf. [24, 9, 13, 25]. We ten introduce a modified velocity space wic represents te essential novelty of our discretization. Compared to te previously introduced space tis modified velocity space is more involved. Details on te construction and realization of tis space are separately treated in te subsequent subsection 3. We also discuss an alternative approac wic allows to circumvent te construction of tis space in subsection 3.6. 2.2.1. H(div, Ω-conforming finite elements: te BDM k space. Altoug te velocity solution of te Stokes problem will typically be at least H 1 (Ω-regular, we do not consider H 1 (Ω-conforming finite elements. Instead, we base our discretization on (only H(div, Ω-conforming, i.e. normal-continuous, finite elements. ese are very well-suited for a stable discretization of te incompressibility constraint [24, 11, 25]. Note tat te non-conformity wit respect to H 1 will be dealt wit using Discontinuous Galerkin formulations in te discretization. We recall te definition of H(div, Ω: H(div, Ω := {v [L 2 (Ω] d : div(v L 2 (Ω}. We use piecewise (vector-valued polynomials for te approximation of te velocity, so tat on eac element te functions are automatically in H(div,,. or global conformity, continuity of te normal component is necessary. We consider te well-known BDM k space: W := {u [P k ( ] d : [u n] =, } H(div, Ω. (2.2 2.2.2. angential facet unknowns. e space W is not H 1 -conforming so tat tangential continuity as to be imposed weakly troug a DG formulation for te viscosity terms, for instance as in [11]. However, te incorporation of tangential continuity in usual DG scemes leads to a uge amount of couplings between neigboring elements wic increases te costs for solving te linear systems. o approac tis problem, we decouple element unknowns by introducing additional unknowns on te facets troug wic tangential continuity is implemented weakly. We note tat suc a mecanism is not necessary for te normal direction as normal-continuity is implemented (strongly in te space W. We introduce te space for te facet unknowns := {u [P k 1 ( ] d : u n =, int }, (2.3 wic can be seen as an approximation to te tangential trace of te velocity on te facets. Note tat we only consider polynomials up to degree k 1 in wereas we ave order k polynomials in W. urter, functions in ave normal component zero. 4
2.2.3. e pressure space. or te pressure, te appropriate finite element space to te velocity space W is te space of piecewise polynomials wic are discontinuous and of one degree less: Q := P k 1 (. (2.4 is velocity-pressure pair W /Q fulfills div(w = Q and we ence ave tat if a velocity u W is weakly incompressible, it is also strongly incompressible: div(u q dx = q Q div(u = in Ω. (2.5 Ω Hence, tis coice of te velocity-pressure pair easily results in a pointwise divergence free solution. is property is crucial to obtain energy-stable scemes for Navier- Stokes discretizations, cf. [1]. We furter note, tat tis finite element pair furter as te remarkable property of providing an LBB-constant tat is robust in te mes size and te polynomial degree k, cf. Lemma 4.4 and [22]. 2.2.4. A modified BDM k space. Wit te velocity unknowns in W and we ave polynomials up to degree k associated to te normal direction of a facet, but only polynomials up to degree k 1 for te tangential direction of a facet. As tese unknowns are te globally coupled unknowns, tey essentially decide on te computational costs associated wit solving linear systems. e question arises if we can reduce te polynomial degree for te normal direction also to degree k 1. We can acieve tis only by relaxing te normal continuity of W. o tis end, we introduce a modified inite Element space wit relaxed H(div-conformity, W := {u [P k ( ] d : Π k 1 [u n] =, } H(div, Ω, (2.6 were Π k 1 : [L 2 ( ] d [P k 1 ( ] d is te L 2 ( projection into P k 1 ( : ( Π k 1 w v ds = w v ds v [P k 1 ( ] d. (2.7 Details on te constructions of te finite element space W are given below, in section 3. unctions in W are only almost normal-continuous, but can be normaldiscontinuous in te igest orders. e relaxation reduces te number of unknowns tat are sared by neigboring elements wic improves te sparsity pattern of te linear systems. However, tis modification comes at a prize. We obviously ave W W and W = W,, but te pair W /Q only as te local property div(w = div(w = Q. If a velocity u W is weakly incompressible (in te sense of te left and side of (2.5, we only ave div(u = in and Π k 1 [u ] =,. (2.8 In order to obtain mass conservative velocity fields we are missing normal continuity in te iger order moments, (id Π k 1 [u ] =. or te discretization of te velocity field we use te composite space U := W. (2.9 5
and define te tangential jump operator [u t ] = u t u, int,. We note tat te jump [[u t ] is element-sided tat means tat it can take different values for different sides of te same facet. urter, we note tat for ext we ave u = so tat [u t ] = u t. A suitable discrete norm on U wic mimics te H 1 (Ω norm and a suitable norm for te velocity pressure space U Q are u 2 1 := { u 2 + 1 } Πk 1 [u t ] 2, (u, p 1 := ν u 1 + 1 p L2 (Ω. ν (2.1 2.3. Relaxed H(div-conforming HDG formulation. e basis of our Hybrid DG formulation is te formulation in [25] wic is based on te spaces W, Q and. In contrast to tat formulation, we now replace W wit W. is will lead to an order-optimal basic metod wit a reduced number of globally coupled unknowns. Due to te relaxation of te normal-continuity velocity solutions will (in contrast to te discretization in [25] wit W be neiter exactly solenoidal nor pressure robust, i.e. te error in te velocity solution will depend on te approximation of te pressure. However, bot drawbacks can be dealt wit using a simple and computationally ceap reconstruction operator. In section 2.4, we address te issue of restoring normal-continuity strongly (after te solution of linear systems by a reconstruction operator. In section 2.5, we discuss ow to also restore pressure robustness. e basic discretization is as follows: ind u = (u, u U and p Q, s.t. { A (u, v + B (v, p = f(v v U, (B B (u, q = q Q, wit te bilinear forms corresponding to viscosity (A, pressure and incompressibility (B defined in te following. or te viscosity we introduce te bilinear form A for u, v U as A (u, v := ν u : v dx ν u n Πk 1 [v t ] ds (2.11 ν v n Πk 1 [u t ] ds + ν λk2 Πk 1 [u t ] Π k 1 [v t ] ds, were λ is cosen, suc tat te bilinearform is coercive w.r.t. 1 on U, see Lemma 4.3 below. e L 2 ( -projection Π k 1, c.f. (2.7, realizes a reduced stabilization [25, Section 2.2.1]. If we replace Π k 1 wit id in (2.11 we obtain te usual ybridized interior penalty formulation of te viscosity as used in [24, 3]. In te literature of DG metods many alternatives to te interior penalty metod are known. or many of tese alternatives tere is a corresponding HDG version wic is also applicable, we owever restrict to te interior penalty metod for ease of presentation. Implementational aspects of te projection operator Π k 1 are discussed below, in section 3.5. e bilinear form for te pressure part and te incompressibility constraint is B (u, p := p div(u dx for u U, p Q. (2.12 We denote te discretization in (B as our basic discretization. In te next subsections we introduce modifications related to te relaxed H(div-conformity. 6
2.4. A reconstruction operator for strong H(div-conformity. Velocity solutions to (B are in general not solenoidal as (id Π k 1 [u n] can be nonzero. In Navier-Stokes or oter coupled problems were te velocity solution serves as an advective velocity it is benefitial to ave an exact solenoidal field. Similarly to te approac in [1] we propose to use a reconstruction operator for te velocity solution wic restores H(div-conformity. We denote suc a reconstruction operator as R W : W W and define te canonical extension to W as R U : W W, R U u := (R W u, u. (2.13 and R U, re- We make te following assumptions on tis reconstuction operator: Assumption 1. We assume tat te reconstruction operators R W spectively, fulfill te following conditions: R W v W for all v W, (2.14a (R W v, ϕ L2 ( = (v, ϕ L2 ( for all ϕ P k 2 (, v W,, (2.14b R U v 1 v 1 for all v U. (2.14c Here, (2.14a ensures H(div-conformity, (2.14b describes a constistency property in L 2 (Ω and (2.14c ensures stability in te discrete energy norm. A reconstruction operator wit tese properties maps weakly divergence free velocities (in te sense of (B onto pointwise divergence free velocities, cf. Lemma 4.8 below. One possible coice is a (Discontinuous Galerkin generalization of te classical BDM interpolation [5, Proposition 2.3.2], R BDM W : [H 1 ( ] d W. It as also been used in [15]. e interpolation is defined element-by-element for u [H 1 ( ] d by R BDM W u n ϕ ds = {u } nϕ ds for all ϕ P k (,, (2.15a R BDM W u ϕ dx = u ϕ ds for all ϕ N k 2 (, (2.15b wit N k 2 ( := [P k 2 ( ] d + [P k 2 ( ] d x. We note tat te definition of te DG-version BDM interpolation is sligtly different from te one used in [1]. We note tat tis reconstruction operator satisfies all te previously mentioned assumptions. e assumptions (2.14a and (2.14b are satisfied by construction and stability in te sense of (2.14c as also been sown already in [1]. e reconstruction operator in [1] is designed for a fully discontinuous velocity space. Based on te finite element basis tat we discuss in section 3 we propose a simple and efficient realization of a similar, but different, interpolation operator (see section 3.4 wic also fulfills assumption 1. 2.5. Pressure robust relaxed H(div-conforming HDG formulation. It is well known tat irrotational parts (in te sense of a continuous Helmoltz decomposition of an exterior force f are L 2 -ortogonal on exactly divergence free velocities v [H 1 (Ω] d. Neverteless tis ortogonality may not old true in te discrete case leading to a velocity error estimate wic depends on te pressure approximation [27]. Recent works [7, 23, 26, 28] considering different velocity and pressure spaces ave sown tat a modification of te rigt and side allows to obtain pressure-robust, tus pressure independent, error estimates. is is acieved by mapping weakly divergence free test functions to exactly divergence free functions resulting in a restored L 2 ortogonality wit respect to te irrotational parts of f. Note tat H(div-conforming 7
metods as for example [25, 11] provide exactly divergence free velocities and tus do not suffer from te described problems. Due to te modification (2.6 functions u U are only weakly divergence free, see equation (2.8. e reconstruction operator introduced in te Section 2.4 allows to obtain a pressure robust discretization also for our new metod. e discretization ten takes te form: ind u U and p Q, s.t. { A (u, v + B (v, p = f(r U v v U, B (u, q = q Q. (PR Solutions u U to (PR are not exactly divergence free but provide a pressure independent velocity error estimate. is involves a Strang type consistency estimate and is proven in section 4.3. urtermore, solutions to (PR can be post-processed wit a subsequent application of R U to obtain an exactly divergence free solution. 3. On te construction of te finite element spaces. In tis section we address te construction of te finite element spaces W,, an efficient reconstruction operator R W and te realization of te projected jumps operator Π k 1. In te subsections 3.1 and 3.2 we introduce te finite element basis functions on a reference element and explain ow tese are composed to a global finite element space W in subsection 3.3. Based on tese preparations we can introduce an efficient reconstruction operator R W wic meets te requirements of Assumption 1 in subsection 3.4. In section 3.5 we explain ow a ierarcical decomposition of basis functions for can be used to realize te projected jumps operator Π k 1. We conclude te section wit subsection 3.6 were we discuss an equivalent formulation based on (simpler scalar finite element spaces. 3.1. Construction of a local H(div-conforming E Space. In tis section we define sape functions to construct a basis of a local space Ŵ on te reference triangle given as te convex ull of te vertices V = {V i } 3 i=1 := {( 1,, (1,, (, 1} and V = {V i } 4 i=1 := {( 1,,, (1,,, (, 1,, (,, 1} for two and tree dimensions respectively. Using a proper transformation we can ten construct te global spaces W and W, see section 3.3. or te construction of te local space Ŵ we refer to [3] were te basic concepts presented in [35] are combined wit some adaptations leading to a sparsity optimized ig order basis. e novel coice of te element basis functions is suc tat teir normal component form a ierarcical L 2 ortogonal basis on faces, see Lemma 3.1. e idea in [35] is to construct conforming finite element spaces for H 1, H(div, H(curl and L 2 wic fit in te setting of te exact de Ram Complex. is construction allows to split te space W H(div into divergence free basis functions given by te curl of basis functions in S H(curl and functions wit a non zero divergence. On te reference triangle tis leads to te following decomposition Ŵ = R Ŵ Ŵ divfree, Ŵ div,, (3.1 wit te lowest order Raviart-omas subspace R, see [29], te subspace of iger order divergence free facet functions Ŵ te subspace of cell based divergence free basis functions Ŵ divfree, and te subspace of cell based functions tat ave a non zero divergence Ŵ div,. In igure 3.1 a simple sketc of te decomposition in te two dimensional case is presented. 8
ig. 3.1. Sketc of velocity fields in te decomposition of te space Ŵ into te sub spaces (from left to rigt R, Ŵ, Ŵ divfree, and Ŵ div,, in te two dimensional case. Now let p (α,β n be te n-t Jacobi polynomial and ˆp (α,β n polynomial (see [1, 2, 4], tus p (α,β n (x:= ˆp (α,β n (x := 1 d n ( (1 x α 2 n n!(1 x α (1 + x β dx n (1+x β (x 2 1 n, x 1 te n-t integrated Jacobi n N, α, β > 1, p (α,β n 1 (ξ dξ, n 1, ˆpα (x = 1. (x = p α n(x and In our case we use β = so we skip to te simpler notation p (α, n (x = ˆp α n(x. Note tat tere olds te following ortogonality property. ˆp (α, n 1 1 1 1 (1 x α p α j (xp α l (x dx = c α j δ jl wit c α j = 2 α+1 2j + α + 1, (3.2 (1 x α ˆp α j (xˆp α l (x dx = for j l > 2. (3.3 urtermore we define λ i P 1 ( as te barycentric coordinates uniquely determined by λ i (V j = δ ij. 3.1.1. Basis for two dimensions. or a fixed order k we use te same construction of te basis as presented in capter( 3 in [3]. Let [f 1, f 2 ] be te edge running from vertex V f1 and V f2 and define u i := ˆp λ 2 λ 1 i λ 2+λ 1 (λ 2 + λ 1 i and v ij := ˆp 2i 1 j (2λ 3 1. en we ave: e lowest order Raviart omas basis functions φ [e1,e2] := curl(λ e1 λ e2 λ e1 curl(λ e2 suc tat R = span({φ [1,2], φ [2,3], φ [3,1] }. Hig-order-edge-based basis functions φ [f1,f2] i and Φ [f1,f1] := {φ [e1,e2] i } for 1 i k, suc tat := curl Ŵ = span(φ [1,2] span(φ [2,3] span(φ [3,1]. ( ( ˆp λf2 λ f1 i+1 λ f2 +λ f1 (λ f2 + λ f1 i+1 Divergence free cell based basis functions φ (a ij := curl (u i v ij wit Φ (a := } for i 2, j 1, i + j k + 1, suc tat {φ (a ij Ŵ divfree, = span(φ (a. 9
Cell based basis functions wit a non zero divergence φ (b 1l := 2φ [1,2] ˆp 3 l (2λ3 1 and φ (c ij := curl(u i v ij wit Φ (b := {φ (b 1l } for 1 l k 1 and Φ(c := {φ (c ij } for i 2, j 1, i + j k + 1, suc tat Ŵ div, = span(φ (b span(φ (c. 3.1.2. Basis for tree dimensions. Let k be a fixed order. or our local basis we nearly use te same construction as presented in capter 4 in [3]. Let = [f 1, f 2, f 3 ] be te face defined as te convex ull of V f1, V f2 and V f3 and define u i := ˆp i v ij := ˆp 2i 1 j ( λ2 λ 1 (λ 2 + λ 1 i, w ijk := ˆp 2i+2j 2 λ 2 + λ 1 ( 2λ3 (1 λ 4 1 λ 4 (1 λ 4 j, j (2λ 4 1, and u i := ˆp i v ij := ˆp 2i 1 j ( λf2 λ f1 λ f2 + λ f1 ( λf3 λ f2 λ f1 (λ f2 + λ f1 i, N [f1,f2] := λ f1 λ f2 λ f1 λ f2 λ f3 + λ f2 + λ f1 (λ f3 + λ f2 + λ f1 j, were N [f1,f2] is te lowest order Nedéléc function for te edge [f 1, f 2 ]. en we ave: e lowest order Raviart omas basis functions φ [f1,f2,f3] := λ f1 λ f2 λ f3 + λ f2 λ f3 λ f1 + λ f3 λ f1 λ f2, suc tat R = span(φ wit Φ := {φ [1,2,3], φ [1,3,4], φ [1,2,4], φ [2,3,4] } Hig order face based basis functions φ l := (N [f1,f2] v ( 2l and φ ij := u i+1 v (i+1(j+1 = u i+1 v (i+1(j+1 wit Φ := {φ l } {φ ij } for 1 l k and i 1, i + j k, suc tat Ŵ = span(φ [1,2,3] span(φ [1,3,4] span(φ [1,2,4] span(φ [2,3,4]. divergence free cell based basis functions φ (a 1jl := (N [1,2] v 2j w 2jl and φ (b ijl := ( u iv ij w ijl and φ (c ijl := ( (u iv ij w ijl wit Φ (a := {φ (a 1jl } for j, l 1, j + l k, and Φ (b := {φ (b ijl } for i 2, j, l 1, 1 + j + l k + 2, and Φ (c := {φ (c ijl } for i 2 and j, l 1, i + j + l k + 2, suc tat Ŵ divfree, = span(φ (a span(φ (b span(φ (c. Cell based basis functions wit a non zero divergence φ (d 1l := 4φ[1,2,3] φ (e [1,2] 1jl := 2N w 2jl v 2j, and φ (f ijl for 1 l k 1 and Φ (e := {φ (e for i 2, j, l 1, i + j + l k + 2, suc tat w 21l, and := w ijl u i v ij wit Φ (d := {φ (d 1l } 1jl } for j, l 1, j + l k, and Φ(f := {φ (f ijl } Ŵ div, = span(φ (d span(φ (e span(φ (f. 1
Remark 1. Due to te decomposition (3.1 we can conclude tat for an arbitrary function v in Ŵ wit div(v = te coefficients corresponding to te basis functions of te subspace Ŵ div,, wic ave a non zero divergence, are equal to zero. We can use tis for te approximation of te saddle point problem (B keeping in mind te pressure variable can be interpreted as a Langrian multiplier to fulfill te incompressibility constraint div(u =. Were te lowest order piece wise constant pressure basis functions compensate te divergence of te Raviart omas basis functions of te velocity, te ig order part of te pressure space is needed to andle te divergence of velocity basis functions in Ŵ div,. In order to reduce te size of te problem one can now simply remove te basis functions of Ŵ div, and use te pressure space Q := P ( instead. Note tat tis as no influence in te (optimal error estimation of te velocity and furtermore one can use a simple element-wise post processing to obtain a ig order approximation of te pressure. Remark 2. e differences of te presented basis compared to te one given in [3] are te ig order face based basis functions φ l were we ave cosen a different index and a proper scaling into te direction of te opposite vertex of te face function vij. Note tat tis as no influence on te linear in dependency, tus te presented set of basis functions is still a basis for Ŵ. urtermore we still ave div(φ l =, tus we ave te same sparsity pattern of te div div-stiffness matrix and even a better sparsity pattern of te mass matrix (see Lemma 3.2 as in [3]. 3.2. Properties of te local basis. Beside te sparsity properties of te stiffness and mass matrix (see [3] for te basis introduced in section 3.1.1 and 3.1.2 in tis section, we proof properties tat are important for proving Assumption 1 for te reconstruction operator tat we present below, in section 3.4. Lemma 3.1 (L 2 -normal ortogonality. e basis presented in section 3.1 as a L 2 -ortogonal normal trace. In two dimensions tere olds (c=c(i, j> (φ i n(φ j n ds = cδ ij i, j =,.., k. (3.4a In tree dimensions tere olds (c = c(i, j, l, m > (φ ij n(φ lm n ds = cδ il δ jm, i, j, l, m k, i+j, l+m k,. (3.4b Proof. We start wit te two dimensional case and ( te( lower edge = [1, 2]. or i, j 1 te basis functions are given by φ [1,2] i = curl ˆp λ 2 λ 1 i+1 λ 2+λ 1 (λ 2 + λ 1. i+1 In two dimensions te curl times te normal vector equals te tangential derivative, tus for te partial derivation wit respect to x (φ [1,2] i n = x ˆp i+1(x = p i (x, were we used tat λ 2 + λ 1 = 1 and λ 2 λ 1 = x on. Next note tat for i, j = te normal component of te lowest order Raviart omas basis function φ is constant on tus equivalent to p. Using property (3.2 it follows 1 (φ [1,2] i n(φ [1,2] j n ds = p i (xp j(x dx = c i δ ij 1 for i, j k 11
e oter two edges follow analogue. or te proof in tree dimensions let = [1, 2, 3] and start wit i 1. e normal vector on is given by n = (,, 1, so we ave φ ij n =( u i+1 v(i+1(j+1 ( n x = x (ˆp i+1 (1 y i+1 y ˆp 2i+1 j+1 (2y 1 1 y ( x y (ˆp i+1 (1 y i+1 x ˆp 2i+1 j+1 (2y 1 1 y ( x =p i (1 y i p 2i+1 j (2y 1 2. 1 y Using a Duffy like transformation from D 1 : ( 1, 1 (, 1, wit D 1 (ˆx, ŷ = (ˆx(1 ŷ, ŷ we get (φ ij n(φ lm n ds 1 = 4 = 4 1 1 1 1 p i (ˆx (1 ŷ i p 2i+1 j (2ŷ 1 p l (ˆx (1 ŷ l p 2l+1 m (2ŷ 1 (1 ŷ dŷ dˆx p i (ˆx p l (ˆx dˆx 1 p 2i+1 j (2ŷ 1 p 2l+1 m (2ŷ 1 (1 ŷ i+l+1 dŷ Using a transformation D 2 : ( 1, 1 (, 1 for te second integral and (3.2 we see tat te first integral vanises for i j and te second integral for i = j, and so (φ i,j n(φ l,m n ds = c(i, j, l, mδ il δ jm i, l 1, i + j, l + m k. or i = l = we ave wit N [f1,f2] = 1 2 (y + z 1, x, x, tus on as z =, ( φ j n = (N [f1,f2] v2j n = x x 2 ˆp3 j(2y 1 ( y 1 + y 2 = ˆp 3 j (2y 1 (1 yp 3 j 1 (2y 1. Using te Duffy like transformation D 1 and D 2 we ave similar as before (φ j n(φ m n ds = 2 1 1 (1 ŷˆp 3 j(ŷˆp 3 m(ŷ (1 ŷ2 2 ˆp 3 j(2y 1 (ˆp 3 j (ŷp 3 m 1(ŷ + ˆp 3 m(ŷp 3 j 1(ŷ (1 ŷ3 p 3 4 j 1(ŷp 3 m 1(ŷ dŷ. e integral of te last term is equal to δ jm due to property (3.2. or te first of te mixed terms we get wit integration by parts and ˆp 3 j ( 1 = 1 (1 ŷ 2 1 ( ˆp 3 1 2 j(ŷp 3 m 1(ŷ dŷ = (1 ŷˆp 3 j(ŷ + 1 Putting all terms togeter we ave (φ j n(φ m n ds = c(j, mδ jm 1 j, m k 12 (1 ŷ2 p 3 2 j 1(ŷ ˆp 3 m(ŷ dŷ,
e ortogonality between functions wit wit i = and i 1 follows similar by using te Duffy like transformation and (3.2. e ortogonality wit respect to te lowest order Raviart omas functions φ follows wit te same arguments as in te two dimensional case. e oter faces follow analogue. Lemma 3.2. e igest order facet basis functions from section 3.1 are L 2 ( ˆ - ortogonal on polynomials up to degree k 2. ere olds φ q dx = q [P k 2 ( ] d,. for φ = φ k (in two dimensions and φ {φ k ij } i+j=k (in tree dimensions. Proof. We start wit te two dimensional case and te lower edge = [1, 2]. e igest order edge basis functions is given by φ [1,2] k = curl(ˆp k+1 ( x 1 y (1 yk+1, tus ( φ [1,2] k = p x k( 1 y x(1 yk 1 ˆp x k+1( 1 y (k + 1(1 yk, p x k( (1 yk. 1 y Using a monomial basis for P k 2 (, so x m y n wit m + n k 2, we get for te first component (φ k 1 x m y n d(x, y ( = p x k( 1 y x(1 yk 1 ˆp x k+1( (k + 1(1 yk x m y n d(x, y. 1 y Wit te Duffy like transformation D 1 as in te proof of Lemma 3.1 we get (φ k 1 x m y n d(x, y = = 1 1 1 1 1 p k(ˆxˆx m+1 (1 ŷ k ŷ n (k + 1ˆp k+1(ˆxˆx m (1 ŷ k 1 ŷ n dŷ dˆx p k(ˆxˆx m+1 dˆx 1 (1 ŷ k ŷ n dŷ 1 (k + 1 ˆp k+1(ˆxˆx m dˆx 1 1 (1 ŷ k 1 ŷ n dŷ =, wat follows due to i k 2 and te ortogonality properties for te Legendre and te integrated Legendre polynomials, equation (3.2 and (3.3. or te second component we proceed similar. or te proof in te tree dimensional case let = [1, 2, 3] and start wit i 1 suc tat i + j = k. e first component of φ ij = u i+1 v(i+1(j+1 is given by ( φ ij 1 = y u (i+1 zv(i+1(j+1 zu (i+1 yv(i+1(j+1 = ac bc, wit y u (i+1 = zu (i+1 = a b and c = zv(i+1(j+1 yv(i+1(j+1, tus ( a := p x ( i x(1 y z i 1 b := ˆp x i+1 (i + 1(1 y z i 1 y z 1 y z ( ( 2y + z 1 ( 2y + z 1 c := p 2i+1 j 2y(1 z j 1 ˆp 2i+1 j+1 (j + 1(1 z j 1 z 1 z ( 2y + z 1 p 2i+1 j 2(1 z j. 1 z 13
Using a Duffy like transformation D 3 : ( 1, 1 (, 1 (, 1, wit D 3 (ˆx, ŷ, ẑ = (ˆx(1 ŷ(1 ẑ, ŷ(1 ẑ, ẑ we get for te integral of ac multiplied wit a monome x m y n z r wit m + n + r k 2 ac x m y n z r d(x, y, z = 1 1 p i (ˆxˆx m+1 dˆx 1 1 (1 ẑ i+j+m+n+2 ẑ r dẑ (1 ŷ i+m+1 ŷ n ( 2(1 ŷp 2i+1 j (2ŷ 1 + (j + 1ˆp 2i+1 j+1 dŷ. or m i 2 te integral wit respect to ˆx vanises, so it remains te case m > i 2 i m + 1. or te integral wit respect to ŷ we get using integration by parts 1 1 = = (1 ŷ i+m+1 ŷ n ( 2(1 ŷp 2i+1 j (2ŷ 1 + (j + 1ˆp 2i+1 j+1 dŷ 1 2(1 ŷ i+m+2 ŷ n p 2i+1 j (2ŷ 1 dŷ 2(1 ŷ i+m+2 ŷ n p 2i+1 j (2ŷ 1 dŷ j + 1 i + l + 2 1 1 (j + 1(1 ŷ i+m+1 ŷ n ˆp 2i+1 j+1 dŷ (1 ŷ i+m+2 (2ŷ n p 2i+1 j (2ŷ 1 + ˆp 2i+1 j+1 (2ŷ 1nŷn 1 dŷ. Now as 2i + 1 i + m + 2 and (i + m + 2 (2i + 1 + n j 1 we can write ŷ n (1 ŷ i+m+2 = (1 ŷ 2i+1 w(ŷ were w is polynomial of order less or equal j 1. By tis te integrals of te jacobian polynomials vanis due to (3.2, and wit a similar argument also te integral of te integrated Jacobian polynomial vanis due to (3.3. Using te same tecniques one sows tat also te integral bc x m y n z r d(x, y, z vanises wat leads to te ortogonality of te first component (φ ij 1. In a similar way one proves te ortogonality also for te oter two components. It remains te proof for te face based basis functions wit i =, tus φ [f1,f2] k := (N v2k. We start wit te second component (φ k 2 = z (N [f1,f2] v 2k 1 x (N [f1,f2] v 2k 3 = ˆp 3 k( 2y + z 1 1 z (1 z k + (y + z 1p 3 k 1( 2y + z 1 y(1 z k 2 1 z k 2 (y + z 1ˆp3 k( 2y + z 1 (1 z k 1. 1 z Again using te Duffy like transformation D 3 and te monome x m y n z r wit m + n + r k 2 we get 1 1 (φ k 2 x m y n z r d(x, y, z = ˆx m dˆx (1 ẑ k+m+n+2 ẑ r dẑ 1 ( 1 1 ŷ(1 ŷ m+2 ŷ n p 3 k 1(2ŷ 1 dŷ + (1 ŷ m+1 ŷ n ˆp 3 k(2ŷ 1 dŷ 14 k 2 1 (1 ŷ m+2 ŷ n ˆp 3 k(2ŷ 1 dŷ.
or m = 1 te integral wit respect to ˆx vanises, and for m > 1 all integrals wit respect to ŷ vanis due to (3.2 and (3.3. or m = we proceed similar as for te proof of te first type of face based basis functions. irst apply integration by parts for te second integral wit respect to ŷ 1 (1 ŷŷ n ˆp 3 k(2ŷ 1 dŷ = 1 Adding up all integrals ten gives... = k 2 1 1 (1 ŷ 2 ŷ(1 ŷ 2 ŷ n p 3 k 1(2ŷ 1 dŷ + (1 ŷ 2 ŷ n ˆp 3 k(2ŷ 1 dŷ + n 2 2 ( nŷ n 1 ˆp 3 k(2ŷ 1 + ŷ n p 3 k 1(2ŷ 1 dŷ. 1 1 (1 ŷ 2 ŷ n p 3 k 1(2ŷ 1 dŷ (1 ŷ 2 ŷ n 1 ˆp 3 k(2ŷ 1 dŷ. e first two integrals can be summed up leading to a coefficient (1 ŷ 3, tus vanis due to (3.2. or te oter two terms one uses integration by parts once again and uses (3.3 to finally conclude te ortogonality of te second component. or te oter two components (φ k 1 and (φ k 3 one proceeds similar, wat finises te proof. 3.3. Construction of te global E Spaces. As usual wit p-inite Elements, we associate element basis functions in W (or W wit facets or cells. Cell type basis functions ave zero normal trace wile te normal trace of te basis functions associated wit one facet span te polynomial space up to order k. e novel coice of te element basis functions is suc tat teir normal component form a ierarcical L 2 ortogonal basis on facets. In tree dimensions facet basis functions depend on indices i, j, i.e. φ ij (for te two-dimensional case we remove te index j, i.e. φ i, and we ave span({φ ij n : i + j k} = P k ( and span({φ ij n : i + j = k} P k 1 (. is follows due to te ierarcical structure of integrated Jacobi polynomials and Lemma 3.1 on te reference element. By applying te Piola transformation tis property carries over to te pysical elements in te mes. Remark 3. e normal trace of te Piola mapped basis functions are biortogonal to trivially mapped basis functions, also on curved elements. o obtain te different spaces W and W we use te same local basis functions but put tem togeter differently to define te global functions. or te construction of W we associate all te degrees of freedom to facet basis functions φ ij wit mes facets to obtain normal continuity. or te space W wit relaxed H(div-conformity we only associate degrees of freedom wit i + j < k wit facets. e remaining degrees of freedom wit i + j = k are associated wit cells and are treated as locally (on only one cell supported functions. is means tat te every facet basis functions in W wit i + j = k is split up into two (locally supported basis functions: or a facet and i + j = k te basis function φ ij wit supp(φ ij = is replaced wit te basis functions φ, ij = φ ij and φ, ij = φ ij wic ave supp(φ, ij = and supp(φ, ij =. We note tat te new functions φ, ij and φ, ij (wit non-zero normal trace on can be treated as interior bubble functions. rom tis construction we observe tat a realization of te non-standard space W is not more difficult tan te construction of W. 15
e split-up functions form bases for te local and global spaces W := span({φ, ij : i + j = k},, W := Note tat Lemma 3.2 implies tat φ qdx = for all φ W, for all q Ω W. (3.5 [P k 2 ( ] d. (3.6 3.4. A simple and fast realization of te BDM interpolation. Due to te different association of te degrees of freedom of facet based basis functions (see section 3.3 a discontinuity of te normal component of a function u W only appears in te igest order components. e idea of te reconstruction operator R W is to exploit te origin of tis discontinuity in te basis functions. It simply reverts te break-up of one basis function into two by applying an average on eac facet of te corresponding coefficients. Let be an arbitrary facet wit, suc tat =. In tree dimensions we ave te local representation (in two dimensions we again remove te index j of te normal components (u n = i+j<k (u n = i+j<k α ij φ ij n + i+j=k α ij φ ij n + i+j=k α ij φ, ij n α ij φ, ij n, and its jumps [u n] = (αij α ij φ ij n. i+j=k Here α ij, αij and α ij are given coefficients, and φ, ij, φ, ij are te igest order basis functions associated to te elements and. Note tat φ ij n = φ, ij n = φ, ij n on for i + j = k. Wit α ij = 1 2 (α ij + α ij we define te reconstruction operation R W u suc tat for {, } we ave (R W u n := i+j<k α ij φ ij n + i+j=k α ij φ ij n. (3.7 In te reconstruction we tus only apply tis averaging wic only affects te igest order facet degrees of freedom. Hence, te reconstruction can also be caracterized as a perturbation in te space W. In igure 3.4 we see a sketc of te averaging on one facet of two neigboring elements. Lemma 3.3. e reconstruction operator R W defined by te averaging (3.7 fulfills Assumption 1. Proof. (2.14a follows by construction and (2.14b follow directly from Lemma 3.2. It remains to sow (2.14c. We claim tat for every element tere olds te local estimate R U v 2 v 2 for all v W (, (3.8 16
αij φ, ij α ij φ, ij α ij (φ, ij + φ, ij = α ij φ ij R W ig. 3.2. In te left picture we see two igest order facet functions φ, ij and φ, ij and te continuous low order facet functions of two neigboring triangles and. e reconstruction R W uses te average value of te corresponding coefficients to remove te normal discontinuity in te igest order, see on te rigt. were ( denotes te set of facets surrounding and is te element-local version of 1 wit v 2 := v 2 + 1 (Πk 1 [v t ] 2 ds. o prove (3.8 we first use R U v R U v v + v, (3.9 define w := R U v v and note tat w W {}, cf. (3.5. Wit a transformation to te reference element and equivalence of finite dimensional norms, one easily sows tat on W {} te norms, and, wit v 2, := 1 (id Πk 1 (v n 2 are equivalent wit constants independent of. Hence, R U v 2 w 2, + v 2 = R U v 2 1 v 2 1 + inally, we bound w 2 1, = (id Πk 1 (w n 2 v 2 from wic we conclude (2.14c. 1 w 2,. (id Πk 1 (v n 2 3.5. Projected jumps: Realization. In te definition of te bilinear form A we use te projection into polynomials of degree k 1, Π k 1. We recall ow tis projection can be realized, cf. [25]. Again, we assume an L 2 -ortogonal basis for te facet functions in. is can be constructed from a scalar L 2 -ortogonal Dubiner basis (see [12] for every component on a facet-local coordinate system (spanned by d 1 aligned edges. We denote te tangential space to eac facet by t. Using te ortogonal basis on eac facet, we can (explicitly split te space of (vector-valued polynomials up to degree k, t P k (, into te ortogonal subspaces V k 1 := t P k 1 ( = and V, k 1 := t [P k ( ( P k 1 ( ]. (3.1 We describe ow to compute te element matrix to A on. o tis end we denote by A te bilinear form wic as te same form as A but wit te 17
projection replaced wit te identity. o A we compute te element matrix wit te (temporarilly enriced te local space by adding V, k 1 to te facet space V k 1 = so tat we ave te local facet space V k. We claim tat setting up tis element matrix wit te enriced local space and eliminating te newly added subspace afterwards realizes te implementation of te element matrix to A. We sow tis next. On every facet of we can write every function u t P k ( as te unique sum u = u + λ wit u Vk 1 =, λ V, k 1. We use tis decomposition for trial and test functions in Vk, u = u + λ and v = v + µ, (3.11 wit u, v V k 1 and λ, µ V k 1. e newly introduced test functions µ appear only element-local and only on one facet so tat tis implies A ((u, u, (, µ = wic yields for all µ V k 1 ν u n µ ds+ ν λk2 (ut u λ µ ds = ν λk2 (ut λ µ ds =. (3.12 We see tat λ = (I Π k 1 ut. After elimination of tis new (and only local function we obtain [u t ] = u t u λ = Π k 1 (ut u = Π k 1 (ut u = Π k 1 [u t ]. (3.13 Hence, we ereby obtained a realization of te projection operator. 3.6. An equivalent formulation based on scalar E spaces. We present a different discrete formulation wic results in te same discrete solution but uses te muc simpler finite element spaces for velocity and pressure: V = [P k ( ] d, Λ = [P k 1 ( ] d and Q = P k 1 (. We note tat te velocity spaces are product spaces of scalar-valued finite element spaces for element and facet unknowns witout any continuity between elements. e facet space Λ is responsible for two tasks: e tangential components of te function in Λ take te role of te facet variables in in (B. e normal component is a Lagrangian multiplier introduced to enforce te weak H(div-conformity of solutions. We define te bilinear forms à (u, v := ν u : v dx ν u [v t ] ds (3.14 ν v n Πk 1 [u t ] ds + n Πk 1 ν λk2 Πk 1 [u t ] Π k 1 [v t ] ds, for u = (u, u, v = (v, v V Λ and for u, v V Λ, p Λ B (u, (v, p := p div(u dx + u v n ds. (3.15 18
en te alternative discretization takes te form: ind u V Λ, p Q, s.t. for all v V Λ, q Q tere olds à (u, v + B (v, (u, p + B (u, (v, q = f(v. ( B e conditions B (u, (, q =, q Q and B (u, ((, v n, =, v Λ enforce divergence free solutions witin every element, div(u = and a relaxed normal continuity, Π k 1 [u n] =, respectively. is implies tat velocity solutions to ( B will be in W. We note tat on te subspace of functions in W V we ave tat Ã(, = A (, and B (, = B (, so tat we obtain te same functions u, u t and p as solutions to (B and ( B. e additional unknown u n is an approximation to te normal stress p νn u n as u n is te Lagrangian multiplier for te (weak normal continuity. We note tat altoug te solution to te equivalent formulations (B and ( B are te same, te sparsity pattern is different, cf. [25, Remark 7]. Also, we note tat wile te implementation of te finite element spaces is easier in ( B, te implementation of te projection Π k 1 is still required, cf. te discussion in te previous subsection. urtermore, te usual basis for te simpler space V does not provide te simple and fast realization of te reconstruction operator tat we discussed in section 3.4 so tat te reconstruction and te pressure robust variant (PR will be more involved and computationally more expensive. In [3] a discretization similar to ( B is considered for a Navier-Stokes problem, wit te important difference tat instead of one facet space Λ, two separate facet spaces Λ u and Λp are used were Λu = [Pk ( ] d is te facet space for te momentum conservation in te velocity discretization (and tus no projection is needed wile Λ p = Pk 1 ( implements te weak H(div-conformity. In a similar fasion different discretizations can be derived wen varying different polynomial degrees in Λ u, Λp or Λ. Wen coosing degree k in Λ p and Λu (and removing te projection Π k 1 we obtain yet anoter formulation wit divergence free solutions tat as recently been proposed in [31]. 4. A priori error analysis. In tis section we consider te a priori analysis of te discretizations (B, (PR, bot wit and witout a subsequent application of te reconstruction operator R U tat fulfills Assumption 1. Preliminaries. In order to compare discrete velocity functions u = (u, u U wit functions u U reg := [H 1 (Ω H 2 ( ] d we identify (wit abuse of notation u wit te tuple (u, u for every element, were u is to be understood in te usual trace sense (wic is unique due to te H 1 (Ω regularity. or te purpose of te analysis it is convenient to introduce te big bilinearform for te saddle point problem in (B for (u, p, (v, q (U + U reg L 2 (Ω: K ((u, p, (v, q := A (u, v + B (u, q + B (v, p. (4.1 On top of te discrete norms for u U and p Q, cf. (2.1, we introduce te following stronger norms for u U + U reg and p L 2 (Ω: u 2 1, := u 2 1 + u 2, (u, p 1, := ν u 1, + 1 p L ν 2 (Ω, ese stronger norms 1, allow to control te normal derivatives also for te exact solution. 19
At several occasions in te analysis we use te notation a b for a, b R to express a c b for a constant c tat is independent of. 4.1. Analysis of te basic metod (B. In tis subsection we take a closer look on te analysis of te basic discretization metod (B, cf. section 2.3. e analysis follows standard Strang-type arguments based on consistency, continuity, inf-sup stability (induced by coervitiy of A (, and LBB-stability of B (,. Lemma 4.1 (Consistency. Let (u, p U reg L 2 (Ω be te solution to te Stokes equation (2.1. ere olds for (v, q U Q K ((u, p, (v, q = f v dx E c (u, p, v, (4.2 Ω wit E c (u, p, v := (id Π k 1 u ( ν n + pn (id Πk 1 v ds. (4.3 or (u, p [H 1 (Ω] d [H l ( ] d H l 1 (, l 2 and m = min(k, l 1 we get E c (u, p, v m ( ν u H m+1 ( + 1 ν p Hm ( ν v 1. (4.4 Proof. rom partial integration we get K ((u, p, (v, q = { = { ν u: v dx ν u n Πk 1 ν v n Πk 1 [u t ] ds + }{{} = div(u q dx }{{} = (div( ν u + pv dx } div(v p dx [v t ] ds ν λ Πk 1 [u t ] }{{} = + ν u } n (v Π k 1 [v t ] ds pv n ds = { fv dx + ν u n (vt Π k 1 vt + v t ds ( ν u } n n + p(vn n ds. Π k 1 [v t ] ds On interior facets we ave [ν u n ] = and on boundary facets we ave v t = so tat ν u n vt ds = int [ν u n ] }{{} = 2 v t ds + ext ν u n vt ds =. (4.5 }{{} =
urter we ave, wit σ n = ν u n n + p σ n v n n ds = σ n [v n n] ds + int = ext σ n v n n ds σ n (id Π k 1 (vn n ds. (4.6 In te last step we used tat on te interior facets we ave Π k 1 [v n] = due to v W and on te boundary facets we ave Πk 1 v n =. or te proof of (4.4 we start wit te estimate of te velocity part. Using te Caucy Scwarz inequality and properties of te L 2 projection on one element we get (id Π k 1 u ( ν n ( ν (id Π k 1 + pn (id Πk 1 v ds u + (id Π k 1 p ( ν u H s 1/2 m ( + 1 p H ν m ( (id Π k 1 v 1 2 ν v. Summing over all elements concludes te proof. e pressure estimate follows wit similar tecniques. Lemma 4.2 (Continuity. ere olds A (u, v ν u 1, ν v 1 u U + U reg, v U, (4.7a and B (u, p 1 ν u 1 ν q u U + U reg, p L 2 (Ω, (4.7b wic implies for all (u, p U + U reg L 2 (Ω, (v, q U Q : K ((u, p, (v, q (u, p 1, (v, q 1. (4.8 Proof. Using te Caucy Scwarz inequality on eac triangle we get A (u, v { ν u v + ν u Π k 1 [v t ] + ν v Π k 1 [u t ] + ν λ Πk 1 [u t ] Π k 1 [v t ] }. All terms except te tird term on te rigt and side can naturally be bounded by te element contributions of te norms u 1, and v 1. Wit an inverse inequality for polynomials and Young s inequality we also get a suitable bound for te tird term: v Π k 1 [u t ] v 1 2 Π k 1 [u t ] inally, wit te Caucy Scwarz inequality in R (4.7a is proven. Property (4.7b also follows by simply using te Caucy Scwarz inequality. Lemma 4.3 (Coercivity. ere exists a positive number c λ R suc tat for te stabilization parameter λ > c λ tere olds A (u, u ν u 2 1 u U. 21