FINITE ELEMENT EXTERIOR CALCULUS: FROM HODGE THEORY TO NUMERICAL STABILITY

Size: px
Start display at page:

Download "FINITE ELEMENT EXTERIOR CALCULUS: FROM HODGE THEORY TO NUMERICAL STABILITY"

Transcription

1 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 2, April 2010, Pages S (10) Article electronically publised on January 25, 2010 FINITE ELEMENT EXTERIOR CALCULUS: FROM HODGE THEORY TO NUMERICAL STABILITY DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER Abstract. Tis article reports on te confluence of two streams of researc, one emanating from te fields of numerical analysis and scientific computation, te oter from topology and geometry. In it we consider te numerical discretization of partial differential equations tat are related to differential complexes so tat de Ram coomology and Hodge teory are key tools for exploring te well-posedness of te continuous problem. Te discretization metods we consider are finite element metods, in wic a variational or weak formulation of te PDE problem is approximated by restricting te trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element metods, we develop an abstract Hilbert space framework for analyzing te stability and convergence of suc discretizations. In tis framework, te differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodgeteoretic structures tat ensure well-posedness of te continuous problem from te continuous level to te discrete. We sow stable discretization is acieved if te finite element spaces satisfy two ypoteses: tey can be arranged into a subcomplex of tis Hilbert complex, and tere exists a bounded cocain projection from tat complex to te subcomplex. In te next part of te paper, we consider te most canonical example of te abstract teory, in wic te Hilbert complex is te de Ram complex of a domain in Euclidean space. We use te Koszul complex to construct two families of finite element differential forms, sow tat tese can be arranged in subcomplexes of te de Ram complex in numerous ways, and for eac construct a bounded cocain projection. Te abstract teory terefore applies to give te stability and convergence of finite element approximations of te Hodge Laplacian. Oter applications are considered as well, especially te elasticity complex and its application to te equations of elasticity. Background material is included to make te presentation self-contained for a variety of readers. 1. Introduction Numerical algoritms for te solution of partial differential equations are an essential tool of te modern world. Tey are applied in countless ways every day in problems as varied as te design of aircraft, prediction of climate, development of cardiac devices, and modeling of te financial system. Science, engineering, and Received by te editors June 23, 2009, and, in revised form, August 12, Matematics Subject Classification. Primary: 65N30, 58A14. Key words and prases. Finite element exterior calculus, exterior calculus, de Ram coomology, Hodge teory, Hodge Laplacian, mixed finite elements. Te work of te first autor was supported in part by NSF grant DMS Te work of te second autor was supported in part by NSF grant DMS Te work of te tird autor was supported by te Norwegian Researc Council. 281 c 2010 American Matematical Society

2 282 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER tecnology depend not only on efficient and accurate algoritms for approximating te solutions of a vast and diverse array of differential equations wic arise in applications, but also on matematical analysis of te beavior of computed solutions, in order to validate te computations, determine te ranges of applicability, compare different algoritms, and point te way to improved numerical metods. Given a partial differential equation (PDE) problem, a numerical algoritm approximates te solution by te solution of a finite-dimensional problem wic can be implemented and solved on a computer. Tis discretization depends on a parameter (representing, for example, a grid spacing, mes size, or time step) wic can be adjusted to obtain a more accurate approximate solution, at te cost of a larger finite-dimensional problem. Te matematical analysis of te algoritm aims to describe te relationsip between te true solution and te numerical solution as te discretization parameter is varied. For example, at its most basic, te teory attempts to establis convergence of te discrete solution to te true solution in an appropriate sense as te discretization parameter tends to zero. Underlying te analysis of numerical metods for PDEs is te realization tat convergence depends on consistency and stability of te discretization. Consistency, wose precise definition depends on te particular PDE problem and te type of numerical metod, aims to capture te idea tat te operators and data defining te discrete problem are appropriately close to tose of te true problem for small values of te discretization parameter. Te essence of stability is tat te discrete problem is well-posed, uniformly wit respect to te discretization parameter. Like wellposedness of te PDE problem itself, stability can be very elusive. One migt tink tat well-posedness of te PDE problem, wic means invertibility of te operator, togeter wit consistency of te discretization, would imply invertibility of te discrete operator, since invertible operators between a pair of Banac spaces form an open set in te norm topology. But tis reasoning is bogus. Consistency is not and cannot be defined to mean norm convergence of te discrete operators to te PDE operator, since te PDE operator, being an invertible operator between infinitedimensional spaces, is not compact and so is not te norm limit of finite-dimensional operators. In fact, in te first part of te preceding century, a fundamental, and initially unexpected, realization was made: tat a consistent discretization of a well-posed problem need not be stable [36, 86, 29]. Only for very special classes of problems and algoritms does well-posedness at te continuous level transfer to stability at te discrete level. In oter situations, te development and analysis of stable, consistent algoritms can be a callenging problem, to wic a wide array of matematical tecniques as been applied, just as for establising te well-posedness of PDEs. In tis paper we will consider PDEs tat are related to differential complexes, for wic de Ram coomology and Hodge teory are key tools for exploring te well-posedness of te continuous problem. Tese are linear elliptic PDEs, but tey are a fundamental component of problems arising in many matematical models, including parabolic, yperbolic, and nonlinear problems. Te finite element exterior calculus, wic we develop ere, is a teory wic was developed to capture te key structures of de Ram coomology and Hodge teory at te discrete level and to relate te discrete and continuous structures, in order to obtain stable finite element discretizations.

3 FINITE ELEMENT EXTERIOR CALCULUS Te finite element metod. Te finite element metod, wose development as an approac to te computer solution of PDEs began over 50 years ago and is still flourising today, as proven to be one of te most important tecnologies in numerical PDEs. Finite elements not only provide a metodology to develop numerical algoritms for many different problems, but also a matematical framework in wic to explore teir beavior. Tey are based on a weak or variational form of te PDE problem, and tey fall into te class of Galerkin metods, in wic te trial space for te weak formulation is replaced by a finite-dimensional subspace to obtain te discrete problem. For a finite element metod, tis subspace is a space of piecewise polynomials defined by specifying tree tings: a simplicial decomposition of te domain, and, for eac simplex, a space of polynomials called te sape functions, and a set of degrees of freedom for te sape functions, i.e., a basis for teir dual space, wit eac degree of freedom associated to a face of some dimension of te simplex. Tis allows for efficient implementation of te global piecewise polynomial subspace, wit te degrees of freedom determining te degree of interelement continuity. For readers unfamilar wit te finite element metod, we introduce some basic ideas by considering te approximation of te simplest two-point boundary value problem (1) u (x) =f(x), 1 <x<1, u( 1) = u(1) = 0. Weak solutions to tis problem are sougt in te Sobolev space H 1 ( 1, 1) consisting of functions in L 2 ( 1, 1) wose first derivatives also belong to L 2 ( 1, 1). Indeed, te solution u can be caracterized as te minimizer of te energy functional 1 (2) J(u) := 1 u (x) 2 dx f(x)u(x) dx over te space H 1 ( 1, 1) (wic consists of H 1 ( 1, 1) functions vanising at ±1), or, equivalently, as te solution of te weak problem: Find u H 1 ( 1, 1) suc tat (3) 1 1 u (x)v (x) dx = f(x)v(x) dx, v H 1 ( 1, 1). It is easily seen by integrating by parts tat a smoot solution of te boundary value problem satisfies te weak formulation and tat a solution of te weak formulation wic possesses appropriate smootness will be a solution of te boundary value problem. Letting V denote a finite-dimensional subspace of H 1 ( 1, 1), called te trial space, we may define an approximate solution u V as te minimizer of te functional J over te trial space (te classical Ritz metod), or, equivalently by Galerkin s metod, in wic u V is determined by requiring tat te variation given in te weak problem old only for functions in V, i.e., by te equations 1 1 u (x) v (x) dx = 1 1 f(x) v(x) dx, v V. By coosing a basis for te trial space V, te Galerkin metod reduces to a linear system of algebraic equations for te coefficients of te expansion of u in terms of te basis functions. More specifically, if we write u = M j=1 c jφ j,werete functions φ j form a basis for te trial space, ten te Galerkin equations old if and only if Ac = b, were te coefficient matrix of te linear system is given by

4 284 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER A ij = 1 1 φ j φ i dx and b i = 1 1 fφ i dx. Since tis is a square system of linear equations, it is nonsingular if and only if te only solution for f =0isu =0. Tis follows immediately by coosing v = u. Te simplest finite element metod is obtained by applying Galerkin s metod wit te trial space V consisting of all elements of H 1 ( 1, 1) tat are linear on eac subinterval of some cosen mes 1 = x 0 < x 1 < < x N = 1 of te domain ( 1, 1). Figure 1.1 compares te exact solution u =cos(πx/2) and te finite element solution u in te case of a uniform mes wit N = 14 subintervals. Te derivatives are compared as well. For tis simple problem, u is simply te ortogonal projection of u into V wit respect to te inner product defined by te left-and side of (3), and te finite element metod gives a good approximation even wit a fairly coarse mes. Higer accuracy can easily be obtained by using a finer mes or piecewise polynomials of iger degree. Figure 1.1. Approximation of u = f by te simplest finite element metod. Te left plot sows u and te rigt plot sows u, wit te exact solution in blue and te finite element solution in green. Te weak formulation (3) associated to minimization of te functional (2) is not te only variational formulation tat can be used for discretization, and in more complicated situations oter formulations may bring important advantages. In tis simple situation, we may, for example, start by writing te differential equation u = f as te first-order system σ = u, σ = f. Te pair (σ, u) can ten be caracterized variationally as te unique critical point of te functional 1 I(σ, u) = ( σ2 uσ ) dx + fudx 1 over H 1 ( 1, 1) L 2 ( 1, 1). Equivalently, te pair is te solution of te weak formulation: Find σ H 1 ( 1, 1),u L 2 ( 1, 1) satisfying 1 1 στ dx 1 1 uτ dx =0, τ H 1 ( 1, 1),

5 FINITE ELEMENT EXTERIOR CALCULUS σ vdx= 1 1 fvdx, v L 2 ( 1, 1). Tis is called a mixed formulation of te boundary value problem. Note tat for te mixed formulation, te Diriclet boundary condition is implied by te formulation, as can be seen by integration by parts. Note also tat in tis case te solution is a saddle point of te functional I, not an extremum: I(σ, v) I(σ, u) I(τ,u) for τ H 1 ( 1, 1), v L 2 ( 1, 1). Altoug te mixed formulation is perfectly well-posed, it may easily lead to a discretization wic is not. If we apply Galerkin s metod wit a simple coice of trial subspaces Σ H 1 ( 1, 1) and V L 2 ( 1, 1), we obtain a finite-dimensional linear system, wic, owever, may be singular, or may become increasingly unstable as te mes is refined. Tis concept will be formalized and explored in te next section, but te result of suc instability is clearly visible in simple computations. For example, te coice of continuous piecewise linear functions for bot Σ and V leads to a singular linear system. Te coice of continuous piecewise linear functions for Σ and piecewise constants for V leads to stable discretization and good accuracy. However coosing piecewise quadratics for Σ and piecewise constants for V gives a nonsingular system but unstable approximation (see [25] for furter discussion of tis example). Te dramatic difference between te stable and unstable metods can be seen in Figure 1.2. Figure 1.2. Approximation of te mixed formulation for u = f in one dimension wit two coices of elements, piecewise constants for u and piecewise linears for σ (a stable metod, sown in green), or piecewise constants for u and piecewise quadratics for σ (unstable, sown in red). Te left plot sows u and te rigt plot sows σ, wit te exact solution in blue. (In te rigt plot, te blue curve essentially coincides wit te green curve and ence is not visible.) In one dimension, finding stable pairs of finite-dimensional subspaces for te mixed formulation of te two-point boundary value problem is easy. For any integer r 1, te combination of continuous piecewise polynomials of degree at most r for σ and arbitrary piecewise polynomials of degree at most r 1foru is stable as can be verified via elementary means (and wic can be viewed as a very simple application of te teory presented in tis paper). In iger dimensions, te problem of finding stable combinations of elements is considerably more complicated. Tis is discussed

6 286 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER in Section below. In particular, we sall see tat te coice of continuous piecewise linear functions for σ and piecewise constant functions for u is not stable in more tan one dimension. However, stable element coices are known for tis problem and again may be viewed as a simple application of te finite element exterior calculus developed in tis paper Tecontentsoftispaper.Te brief introduction to te finite element metod just given will be continued in Section 2. In particular, tere we formalize te notions of consistency and stability and establis teir relation to convergence. We sall also give several more computational examples. Wile seemingly simple, some of tese examples may be surprising even to specialists, and tey illustrate te difficulty in obtaining good metods and te need for a teoretical framework in wic to understand suc beaviors. Like te teory of weak solutions of PDEs, te teory of finite element metods is based on functional analysis and takes its most natural form in a Hilbert space setting. In Section 3 of tis paper, we develop an abstract Hilbert space framework, wic captures key elements of Hodge teory and wic can be used to explore te stability of finite element metods. Te most basic object in tis framework is a cocain complex of Hilbert spaces, referred to as a Hilbert complex. Function spaces of suc complexes will occur in te weak formulations of te PDE problems we consider, and te differentials will be differential operators entering into te PDE problem. Te most canonical example of a Hilbert complex is te L 2 de Ram complex of a Riemannian manifold, but it is a far more general object wit oter important realizations. For example, it allows for te definition of spaces of armonic forms and te proof tat tey are isomorpic to te coomology groups. A Hilbert complex includes enoug structure to define an abstract Hodge Laplacian, defined from a variational problem wit a saddle point structure. However, for tese problems to be well-posed, we need te additional property of a closed Hilbert complex, i.e., tat te range of te differentials are closed. In tis framework, te finite element spaces used to compute approximate solutions are represented by finite-dimensional subspaces of te spaces in te closed Hilbert complex. We identify two key properties of tese subspaces: first, tey sould combine to form a subcomplex of te Hilbert complex, and, second, tere sould exist a bounded cocain projection from te Hilbert complex to tis subcomplex. Under tese ypoteses and a minor consistency condition, it is easy to sow tat te subcomplex inerits te coomology of te true complex, i.e., tat te cocain projections induce an isomorpism from te space of armonic forms to te space of discrete armonic forms, and to get an error estimate on te difference between a armonic form and its discrete counterpart. In te applications, tis will be crucial for stable approximation of te PDEs. In fact, a main teme of finite element exterior calculus is tat te same two assumptions, te subcomplex property and te existence of a bounded cocain projection, are te natural ypoteses to establis te stability of te corresponding discrete Hodge Laplacian, defined by te Galerkin metod. In Section 4 we look in more dept at te canonical example of te de Ram complex for a bounded domain in Euclidean space, beginning wit a brief summary of exterior calculus. We interpret te de Ram complex as a Hilbert complex and discuss te PDEs most closely associated wit it. Tis brings us to te topic of Section 5, te construction of finite element de Ram subcomplexes, wic is te

7 FINITE ELEMENT EXTERIOR CALCULUS 287 eart of finite element exterior calculus and te reason for its name. In tis section, we construct finite element spaces of differential forms, i.e., piecewise polynomial spaces defined via a simplicial decomposition and specification of sape functions and degrees of freedom, wic combine to form a subcomplex of te L 2 de Ram complex admitting a bounded cocain projection. First we construct te spaces of polynomial differential forms used for sape functions, relying eavily on te Koszul complex and its properties, and ten we construct te degrees of freedom. We next sow tat te resulting finite element spaces can be efficiently implemented, ave good approximation properties, and can be combined into de Ram subcomplexes. Finally, we construct bounded cocain projections, and, aving verified te ypoteses of te abstract teory, draw conclusions for te finite element approximation of te Hodge Laplacian. In te final two sections of te paper, we make oter applications of te abstract framework. In te last section, we study a differential complex we call te elasticity complex, wic is quite different from te de Ram complex. In particular, one of its differentials is a partial differential operator of second order. Via te finite element exterior calculus of te elasticity complex, we ave obtained te first stable mixed finite elements using polynomial sape functions for te equations of elasticity, wit important applications in solid mecanics Antecedents and related approaces. We now discuss some of te antecedents of finite element exterior calculus and some related approaces. Wile te first compreensive view of finite element exterior calculus, and te first use of tat prase, was in te autors 2006 paper [8], tis was certainly not te first intersection of finite element teory and Hodge teory. In 1957, Witney [88] publised is complex of Witney forms, wic is, in our terminology, a finite element de Ram subcomplex. Witney s goals were geometric. For example, e used tese forms in a proof of de Ram s teorem identifying te coomology of a manifold defined via differential forms (de Ram coomology) wit tat defined via a triangulation and cocains (simplicial coomology). Wit te benefit of indsigt, we may view tis, at least in principle, as an early application of finite elements to reduce te computation of a quantity of interest defined via infinite-dimensional function spaces and operators, to a finite-dimensional computation using piecewise polynomials on a triangulation. Te computed quantities are te Betti numbers of te manifold, i.e., te dimensions of te de Ram coomology spaces. For tese integer quantities, issues of approximation and convergence do not play muc of a role. Te situation is different in te 1976 work of Dodziuk [39] and Dodziuk and Patodi [40], wo considered te approximation of te Hodge Laplacian on a Riemannian manifold by a combinatorial Hodge Laplacian, a sort of finite difference approximation defined on cocains wit respect to a triangulation. Te combinatorial Hodge Laplacian was defined in [39] using te Witney forms, tus realizing te finite difference operator as a sort of finite element approximation. A key result in [39] was a proof of some convergence properties of te Witney forms. In [40] te autors applied tem to sow tat te eigenvalues of te combinatorial Hodge Laplacian converge to tose of te true Hodge Laplacian. Tis is indeed a finite element convergence result, as te autors remark. In 1978, Müller [71] furter developed tis work and used it to prove te Ray Singer conjecture. Tis conjecture equates a topological invariant defined in terms of te Riemannian structure wit one defined in terms of a triangulation and was te original goal of [39, 40].

8 288 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER (Ceeger [30] gave a different, independent proof of te Ray Singer conjecture at about te same time.) Oter spaces of finite element differential forms ave appeared in geometry as well, especially te differential graded algebra of piecewise polynomial forms on a simplicial complex introduced by Sullivan [83, 84]. Baker [15] calls tese Sullivan Witney forms, and, in an early paper bringing finite element analysis tecniques to bear on geometry, gives a numerical analysis of teir accuracy for approximating te eigenvalues of te Hodge Laplacian. Independently of te work of te geometers, during te 1970s and 1980s numerical analysts and computational engineers reinvented various special cases of te Witney forms and developed new variants of tem to use for te solution of partial differential equations on two- and tree-dimensional domains. In tis work, naturally, implementational issues, rates of convergence, and sarp estimates played a more prominent role tan in te geometry literature. Te pioneering paper of Raviart and Tomas [76], presented at a finite element conference in 1975, proposed te first stable finite elements for solving te scalar Laplacian in two dimensions using te mixed formulation. Te mixed formulation involves two unknown fields, te scalar-valued solution, and an additional vector-valued variable representing its gradient. Raviart and Tomas proposed a family of pairs of finite element spaces, one for eac polynomial degree. As was realized later, in te lowest degree case te space tey constructed for te vector-valued variable is just te space of Witney 1-forms, wile tey used piecewise constants, wic are Witney 2-forms, for te scalar variable. For iger degrees, teir elements are te iger-order Witney forms. In tree dimensions, te introduction of Witney 1- and 2-forms for finite element computations and teir iger-degree analogues was made by Nédélec [72] in 1980, wile te polynomial mixed elements wic can be viewed as Sullivan Witney forms were introduced as finite elements by Brezzi, Douglas, and Marini [26] in 1985 in two dimensions, and ten by Nédélec [72] in 1986 in tree dimensions. In 1988 Bossavit, in a paper in te IEE Transactions on Magnetics [21], made te connection between Witney s forms used by geometers and some of te mixed finite element spaces tat ad been proposed for electromagnetics, inspired in part by Kotiuga s P.D. tesis in electrical engineering [66]. Maxwell s equations are naturally formulated in terms of differential forms, and te computational electromagnetics community developed te connection between mixed finite elements and Hodge teory in a number of directions. See, in particular, [17, 37, 57, 58, 59, 70]. Te metods we derive ere are examples of compatible discretization metods, wic means tat at te discrete level tey reproduce, rater tan merely approximate, certain essential structures of te continuous problem. Oter examples of compatible discretization metods for elliptic PDEs are mimetic finite difference metods [16, 27] including covolume metods [74] and te discrete exterior calculus [38]. In tese metods, te fundamental object used to discretize a differential k-form is typically a simplicial cocain; i.e., a number is assigned to eac k-dimensional face of te mes representing te integral of te k-form over te face. Tis is more of a finite difference, rater tan finite element, point of view, recalling te early work of Dodziuk on combinatorial Hodge teory. Since te space of k-dimensional simplicial cocains is isomorpic to te space of Witney k-forms, tere is a close relationsip between tese metods and te simplest metods of te finite element exterior calculus. In some simple cases, te metods even coincide. In contrast to te finite element approac, tese cocain-based approaces do

9 FINITE ELEMENT EXTERIOR CALCULUS 289 not naturally generalize to iger-order metods. Discretizations of exterior calculus and Hodge teory ave also been used for purposes oter tan solving partial differential equations. For example, discrete forms wic are identical or closely related to cocains or te corresponding Witney forms play an important role in geometric modeling, parameterization, and computer grapics. See for example [50, 54, 56, 87] Higligts of te finite element exterior calculus. We close tis introduction by igligting some of te features tat are unique or muc more prominent in te finite element exterior calculus tan in earlier works. We work in an abstract Hilbert space setting tat captures te fundamental structures of te Hodge teory of Riemannian manifolds, but applies more generally. In fact, te paper proceeds in two parts, first te abstract teory for Hilbert complexes, and ten te application to te de Ram complex and Hodge teory and oter applications. Mixed formulations based on saddle point variational principles play a prominent role. In particular, te algoritms we use to approximate te Hodge Laplacian are based on a mixed formulation, as is te analysis of te algoritms. Tis is in contrast to te approac in te geometry literature, were te underlying variational principle is a minimization principle. In te case of te simplest elements, te Witney elements, te two metods are equivalent. Tat is, te discrete solution obtained by te mixed finite element metod using Witney forms, is te same as tat obtained by Dodziuk s combinatorial Laplacian. However, te different viewpoint leads naturally to different approaces to te analysis. Te use of Witney forms for te mixed formulation is obviously a consistent discretization, and te key to te analysis is to establis stability (see te next section for te terminology). However, for te minimization principle, it is unclear weter Witney forms provide a consistent approximation, because tey do not belong to te domain of te exterior coderivative, and, as remarked in [40], tis greatly complicates te analysis. Te results we obtain are bot more easily proven and sarper. Our analysis is based on two main properties of te subspaces used to discretize te Hilbert complex. First, tey can be formed into subcomplexes, wic is a key assumption in muc of te work we ave discussed. Second, tere exist a bounded cocain projection from te Hilbert complex to te subcomplex. Tis is a new feature. In previous work, a cocain projection often played a major role, but it was not bounded, and te existence of bounded cocain projections was not realized. In fact, tey exist quite generally (see Teorem 3.7), and we review te construction for te de Ram complex in Section 5.5. Since we are interested in actual numerical computations, it is important tat our spaces be efficiently implementable. Tis is not true for all piecewise polynomial spaces. As explained in te next section, finite element spaces are a class of piecewise polynomial spaces tat can be implemented efficiently by local computations tanks to te existence of degrees of freedom, and te construction of degrees of freedom and local bases is an important part of te finite element exterior calculus.

10 290 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER For te same reason, ig-order piecewise polynomials are of great importance, and all te constructions and analysis of finite element exterior calculus carries troug for polynomials of arbitrary degree. A prominent aspect of te finite element exterior calculus is te role of two families of spaces of polynomial differential forms, P r Λ k and Pr Λ k, were te index r 1 denotes te polynomial degree and k 0teform degree. Tese are te sape functions for corresponding finite element spaces of differential k-forms wic include, as special cases, te Lagrange finite element family, and most of te stable finite element spaces tat ave been used to define mixed formulations of te Poisson or Maxwell s equations. Te space P1 Λk is te classical space of Witney k-forms. Te finite element spaces based on P r Λ k are te spaces of Sullivan Witney forms. We sow tat for eac polynomial degree r, tereare2 n 1 ways to form tese spaces in de Ram subcomplexes for a domain in n dimensions. Te unified treatment of te spaces P r Λ k and Pr Λ k, particularly teir connections via te Koszul complex, is new to te finite element exterior calculus. Te finite element exterior calculus unifies many of te finite element metods tat ave been developed for solving PDEs arising in fluid and solid mecanics, electromagnetics, and oter areas. Consequently, te metods developed ere ave been widely implemented and applied in scientific and commercial programs suc as GetDP [42], FEniCS [44], EMSolve [45], deal.ii [46], Diffpack [61], Getfem++ [77], and NGSolve [78]. We also note tat, as part of a recent programming effort connected wit te FEniCS project, Logg and Mardal [69] ave implemented te full set of finite element spaces developed in tis paper, strictly following te finite element exterior framework as laid out ere and in [8]. 2. Finite element discretizations In tis section we continue te introduction to te finite element metod begun above. We move beyond te case of one dimension and consider not only te formulation of te metod, but also its analysis. To motivate te teory developed later in tis paper, we present furter examples tat illustrate ow for some problems, even rater simple ones, deriving accurate finite element metods is not a straigtforward process Galerkin metods and finite elements. We consider first a simple problem, wic can be discretized in a straigtforward way, namely te Diriclet problem for Poisson s equation in a polyedral domain R n : (4) u = f in, u =0on. Tis is te generalization to n dimensions of te problem (1) discussed in te introduction, and te solution may again be caracterized as te minimizer of an energy functional analogous to (2) or as te solution of a weak problem analogous to (3). Tis leads to discretization just as for te one-dimensional case, by coosing a trial space V H 1 () and defining te approximate solution u V by Galerkin s metod: grad u (x) grad v(x) dx = f(x)v(x) dx, v V.

11 FINITE ELEMENT EXTERIOR CALCULUS 291 As in one dimension, te simplest finite element metod is obtained by using te trial space consisting of all piecewise linear functions wit respect to a given simplicial triangulation of te domain, wic are continuous and vanis on. A key to te efficacy of tis finite element metod is te existence of a basis for te trial space consisting of functions wic are locally supported, i.e., vanis on all but a small number of te elements of te triangulation. See Figure 2.1. Because of tis, te coefficient matrix of te linear system is easily computed and is sparse, and so te system can be solved efficiently. Figure 2.1. A piecewise linear finite element basis function. More generally, a finite element metod is a Galerkin metod for wic te trial space V is a space of piecewise polynomial functions wic can be obtained by wat is called te finite element assembly process. Tis means tat te space can be defined by specifying te triangulation T and, for eac element T T,aspace of polynomial functions on T called te sape functions, andasetofdegrees of freedom. By degrees of freedom on T, we mean a set of functionals on te space of sape functions, wic can be assigned values arbitrarily to determine a unique sape function. In oter words, te degrees of freedom form a basis for te dual space of te space of sape functions. In te case of piecewise linear finite elements, te sape functions are of course te linear polynomials on T, a space of dimension n + 1, and te degrees of freedom are te n + 1 evaluation functionals p p(x), were x varies over te vertices of T. For te finite element assembly process, we also require tat eac degree of freedom be associated to a face of some dimension of te simplex T. For example, in te case of piecewise linear finite elements, te degree of freedom p p(x) is associated to te vertex x. Given te triangulation, sape functions, and degrees of freedom, te finite element space V is defined as te set of functions on (possibly multivalued on te element boundaries) wose restriction to any T T belongs to te given space of sape functions on T, and for wic te degrees of freedom are single-valued in te sense tat wen two elements sare a common face, te corresponding degrees of freedom take on te same value. Returning again to te example of piecewise linear functions, V is te set of functions wic are linear polynomials on eac element, and wic are single-valued at te vertices. It is easy to see tat tis is precisely te space of continuous piecewise linear functions, wic is a subspace of H 1 (). As anoter example, we could take te sape functions on T to be te polynomials of degree at most 2, and take as degrees of freedom te functions p p(x), x avertexoft, and p pds, e an edge of T. Te resulting assembled finite element space is te e space of all continuous piecewise quadratics. Te finite element assembly process

12 292 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER insures te existence of a computable locally supported basis, wic is important for efficient implementation Consistency, stability, and convergence. We now turn to te important problem of analyzing te error in te finite element metod. To understand wen a Galerkin metod will produce a good approximation to te true solution, we introduce te standard abstract framework. Let V be a Hilbert space, B : V V R a bounded bilinear form, and F : V R a bounded linear form. We assume te problem to be solved can be stated in te form: Find u V suc tat B(u, v) =F (v), v V. Tis problem is called well-posed if for eac F V, tere exists a unique solution u V and te mapping F u is bounded, or, equivalently, if te operator L : V V given by Lu, v = B(u, v) is an isomorpism. For te Diriclet problem for Poisson s equation, (5) V = H 1 (), B(u, v) = grad u(x) grad v(x) dx, F (v) = f(x)v(x) dx. A generalized Galerkin metod for te abstract problem begins wit a finitedimensional normed space V (not necessarily a subspace of V ), a bilinear form B : V V R, and a linear form F : V R, and defines u V by (6) B (u,v)=f (v), v V. A Galerkin metod is te special case of a generalized Galerkin metod for wic V is a subspace of V and te forms B and F are simply te restrictions of te forms B and F to te subspace. Te more general setting is important since it allows te incorporation of additional approximations, suc as numerical integration to evaluate te integrals, and also allows for situations in wic V is not a subspace of V. Altoug we do not treat approximations suc as numerical integration in tis paper, for te fundamental discretization metod tat we study, namely te mixed metod for te abstract Hodge Laplacian introduced in Section 3.4, te trial space V is not a subspace of V, since it involves discrete armonic forms wic will not, in general, belong to te space of armonic forms. Te generalized Galerkin metod (6) may be written L u = F were L : V V is given by L u, v = B (u, v), u, v V. If te finite-dimensional problem is nonsingular, ten we define te norm of te discrete solution operator, L(V,V ), astestability constant of te metod. Of course, in approximating te original problem determined by V, B, andf, by te generalized Galerkin metod given by V, B,andF, we intend tat te space V in some sense approximates V and tat te discrete forms B and F in some sense approximate B and F. Tis is te essence of consistency. Our goal is to prove tat te discrete solution u approximates u in an appropriate sense (convergence). In order to make tese notions precise, we need to compare a function in V to a function in V. To tis end, we suppose tat tere is a restriction operator π : V V,sotatπ u is tougt to be close to u. Ten te consistency error is simply L π u F and te error in te generalized Galerkin metod wic we wis to control is π u u. We immediately get a relation between te error and te consistency error: L 1 π u u = L 1 (L π u F ),

13 FINITE ELEMENT EXTERIOR CALCULUS 293 and so te norm of te error is bounded by te product of te stability constant and te norm of te consistency error: π u u V L 1 L(V,V ) L π u F V. Stated in terms of te bilinear form B, te norm of te consistency error can be written as B (π u, v) F (v) L π u F V = sup. 0 v V v V As for stability, te finite-dimensional problem is nonsingular if and only if B (u, v) γ := inf sup > 0, 0 u V 0 v V u V v V and te stability constant is ten given by γ 1. Often we consider a sequence of spaces V and forms B and F, were we tink of >0 as an index accumulating at 0. Te corresponding generalized Galerkin metod is consistent if te V norm of te consistency error tends to zero wit and it is stable if te stability constant γ 1 is uniformly bounded. For a consistent, stable generalized Galerkin metod, π u u V tends to zero; i.e., te metod is convergent. In te special case of a Galerkin metod, we can bound te consistency error B (π u, v) F (v) B(π u u, v) sup = sup B π u u V. 0 v V v V 0 v V v V In tis case it is natural to coose te restriction π to be te ortogonal projection onto V, and so te consistency error is bounded by te norm of te bilinear form times te error in te best approximation of te solution. Tus we obtain π u u V γ 1 B inf u v V. v V Combining tis wit te triangle inequality, we obtain te basic error estimate for Galerkin metods (7) u u V (1 + γ 1 B ) inf u v V. v V (In fact, in tis Hilbert space setting, te quantity in parenteses can be replaced wit γ 1 B ; see [89].) Note tat a Galerkin metod is consistent as long as te sequence of subspaces V is approximating in V in te sense tat (8) lim inf u v V =0, u V. 0 v V A consistent, stable Galerkin metod converges, and te approximation given by te metod is quasi-optimal; i.e., up to multiplication by a constant, it is as good as te best approximation in te subspace. In practice, it can be quite difficult to sow tat te finite-dimensional problem is nonsingular and to bound te stability constant, but tere is one important case in wic it is easy, namely wen te form B is coercive, i.e., wen tere is a positive constant α for wic B(v, v) α v 2 V, v V, and so γ α. Te bilinear form (5) for Poisson s equation is coercive, as follows from Poincaré s inequality. Tis explains, and can be used to prove, te good convergence beavior of te metod depicted in Figure 1.1.

14 294 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER 2.3. Computational examples Mixed formulation of te Laplacian. For an example of a problem tat fits in te standard abstract framework wit a noncoercive bilinear form, we take te mixed formulation of te Diriclet problem for Poisson s equation, already introduced in one dimension in Section 1.1. Just as tere, we begin by writing Poisson s equation as te first-order system (9) σ = grad u, div σ = f. Te pair (σ, u) can again be caracterized variationally as te unique critical point (a saddle point) of te functional I(σ, u) = ( 1 2 σ σ u div σ) dx + fudx over H(div; ) L 2 (), were H(div; ) = {σ L 2 () : div σ L 2 ()}. Equivalently, it solves te weak problem: Find σ H(div; ),u L 2 () satisfying σ τdx u div τdx=0, τ H(div; ), div σv dx = fvdx, v L 2 (). Tis mixed formulation of Poisson s equation fits in te abstract framework if we define V = H(div; ) L 2 (), B(σ, u; τ,v)= σ τdx u div τdx+ div σv dx, F(τ,v)= fvdx. In tis case te bilinear form B is not coercive, and so te coice of subspaces and te analysis is not so simple as for te standard finite element metod for Poisson s equation, a point we already illustrated in te one-dimensional case. Finite element discretizations based on suc saddle point variational principles are called mixed finite element metods. Tus a mixed finite element for Poisson s equation is obtained by coosing subspaces Σ H(div; ) and V L 2 () and seeking a critical point of I over Σ V. Te resulting Galerkin metod as te form: Find σ Σ,u V satisfying σ τdx u div τdx=0, τ Σ, div σ vdx= fvdx, v V. Tis again reduces to a linear system of algebraic equations. Since te bilinear form is not coercive, it is not automatic tat te linear system is nonsingular, i.e., tat for f = 0, te only solution is σ =0,u = 0. Coosing τ = σ and v = u and adding te discretized variational equations, it follows immediately tat wen f =0,σ = 0. However, u need not vanis unless te condition tat u div τdx=0forallτ Σ implies tat u =0. Inparticular, tis requires tat dim(div Σ ) dim V. Tus, even nonsingularity of te approximate problem depends on a relationsip between te two finite-dimensional spaces. Even if te linear system is nonsingular, tere remains te issue of stability, i.e., of a uniform bound on te inverse operator. As mentioned earlier, te combination of continuous piecewise linear elements for σ and piecewise constants for u is not stable in two dimensions. Te simplest stable elements use te piecewise constants for u, and te lowest-order Raviart-Tomas elements for σ. Tese are finite elements defined wit respect to a triangular

15 FINITE ELEMENT EXTERIOR CALCULUS 295 mes by sape functions of te form (a + bx 1,c+ bx 2 ) and one degree of freedom for eac edge e, namelyσ e σ nds. We sow in Figure 2.2 two numerical computations tat demonstrate te difference between an unstable and a stable coice of elements for tis problem. Te stable metod accurately approximates te true solution u = x(1 x)y(1 y) on(0, 1) (0, 1) wit a piecewise constant, wile te unstable metod is wildly oscillatory Figure 2.2. Approximation of te mixed formulation for Poisson s equation using piecewise constants for u and for σ using eiter continuous piecewise linears (left), or Raviart Tomas elements (rigt). Te plotted quantity is u in eac case. Tis problem is a special case of te Hodge Laplacian wit k = n as discussed briefly in Section 4.2; see especially Section Te error analysis for a variety of finite element metods for tis problem, including te Raviart Tomas elements, is tus a special case of te general teory of tis paper, yielding te error estimates in Section Te vector Laplacian on a nonconvex polygon. Given te subtlety of finding stable pairs of finite element spaces for te mixed variational formulation of Poisson s equation, we migt coose to avoid tis formulation, in favor of te standard formulation, wic leads to a coercive bilinear form. However, wile te standard formulation is easy to discretize for Poisson s equation, additional issues arise already if we try to discretize te vector Poisson equation. For a domain in R 3 wit unit outward normal n, tis is te problem (10) grad div u + curl curl u = f, in, u n =0, curl u n =0, on. Te solution of tis problem can again be caracterized as te minimizer of an appropriate energy functional, (11) J(u) = 1 ( div u 2 + curl u 2 ) dx f udx, 2 but tis time over te space H(curl; ) H(div; ), were H(curl; ) = {u L 2 () curl u L 2 ()} and H(div; ) = {u H(div; ) u n =0on } wit H(div; ) defined above. Tis problem is associated to a coercive bilinear form, but a standard finite element metod based on a trial subspace of te energy space

16 296 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER H(curl; ) H(div; ), e.g., using continuous piecewise linear vector functions, is very problematic. In fact, as we sall illustrate sortly, if te domain is a nonconvex polyedron, for almost all f te Galerkin metod solution will converge to a function tat is not te true solution of te problem! Te essence of tis unfortunate situation is tat any piecewise polynomial subspace of H(curl; ) H(div; ) is a subspace of H 1 () H(div; ), and tis space is a closed subspace of H(curl; ) H(div; ). For a nonconvex polyedron, it is a proper closed subspace and generally te true solution will not belong to it, due to a singularity at te reentrant corner. Tus te metod, wile stable, is inconsistent. For more on tis example, see [35]. An accurate approximation of te vector Poisson equation can be obtained from a mixed finite element formulation, based on te system: σ = div u, grad σ + curl curl u = f in, u n =0, curl u n =0on. Writing tis system in weak form, we obtain te mixed formulation of te problem: find σ H 1 (), u H(curl; ) satisfying στ dx u grad τdx=0, τ H 1 (), grad σ vdx+ curl u curl vdx= f vdx, v H(curl; ). In contrast to a finite element metod based on minimizing te energy (11), a finite element approximation based on te mixed formulation uses separate trial subspaces of H 1 () and H(curl; ), rater tan a single subspace of te intersection H(curl; ) H(div; ). We now illustrate te nonconvergence of a Galerkin metod based on energy minimization and te convergence of one based on te mixed formulation, via computations in two space dimensions (so now te curl of a vector u is te scalar u 2 / x 1 u 1 / x 2 ). For te trial subspaces we make te simplest coices: for te former metod we use continuous piecewise linear functions and for te mixed metod we use continuous piecewise linear functions to approximate σ H 1 () and a variant of te lowest-order Raviart Tomas elements, for wic te sape functions are te infinitesimal rigid motions (a bx 2,c+ bx 1 ) and te degrees of freedom are te tangential moments u u sds for e an edge. Te discrete solutions obtained by te two metods for te problem wen f =( 1, 0) are sown e in Figure 2.3. As we sall sow later in tis paper, te mixed formulation gives an approximation tat provably converges to te true solution, wile, as can be seen from comparing te two plots, te first approximation sceme gives a completely different (and terefore inaccurate) result. Tis problem is again a special case of te Hodge Laplacian, now wit k =1. See Section Te error analysis tus falls witin te teory of tis paper, yielding estimates as in Section Te vector Laplacian on an annulus. In te example just considered, te failure of a standard Galerkin metod based on energy minimization to solve te vector Poisson equation was related to te reentrant corner of te domain and te resulting singular beavior of te solution. A quite different failure mode for tis metod occurs if we take a domain wic is smootly bounded, but not simply connected, e.g., an annulus. In tat case, as discussed below in Section 3.2, te

17 FINITE ELEMENT EXTERIOR CALCULUS 297 Figure 2.3. Approximation of te vector Laplacian by te standard finite element metod (left) and a mixed finite element metod (rigt). Te former metod totally misses te singular beavior of te solution near te reentrant corner. boundary value problem (10) is not well-posed except for special values of te forcing function f. In order to obtain a well-posed problem, te differential equation sould be solved only modulo te space of armonic vector fields (or armonic 1- forms), wic is a one-dimensional space for te annulus, and te solution sould be rendered unique by enforcing ortogonality to te armonic vector fields. If we coose te annulus wit radii 1/2 and 1, and forcing function f =(0,x), te resulting solution, wic can be computed accurately wit a mixed formulation falling witin te teory of tis paper, is displayed on te rigt in Figure 2.4. However, te standard Galerkin metod does not capture te nonuniqueness and computes te discrete solution sown on te left of te same figure, wic is dominated by an approximation of te armonic vector field, and so is noting like te true solution. Figure 2.4. Approximation of te vector Laplacian on an annulus. Te true solution sown ere on te rigt is an (accurate) approximation by a mixed metod. It is ortogonal to te armonic fields and satisfies te differential equation only modulo armonic fields. Te standard Galerkin solution using continuous piecewise linear vector fields, sown on te left, is totally different.

18 298 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER Te Maxwell eigenvalue problem. Anoter situation were a standard finite element metod gives unacceptable results, but a mixed metod succeeds, arises in te approximation of elliptic eigenvalue problems related to te vector Laplacian or Maxwell s equation. Tis will be analyzed in detail later in tis paper, and ere we only present a simple but striking computational example. Consider te eigenvalue problem for te vector Laplacian discussed above, wic we write in mixed form as: find nonzero (σ, u) H 1 () H(curl; ) and λ R satisfying σ τdx grad τ udx=0, τ H 1 (), (12) grad σ vdx+ curl u curl vdx= λ u vdx, v H(curl; ). As explained in Section 3.6.1, tis problem can be split into two subproblems. In particular, if 0 u H(curl; ) and if λ R solves te eigenvalue problem (13) curl u curl vdx= λ u vdx, v H(curl; ), and λ is not equal to zero, ten (σ, u), λ is an eigenpair for (12) wit σ =0. We now consider te solution of te eigenvalue problem (13), wit two different coices of trial subspaces in H(curl; ). Again, to make our point, it is enoug to consider a two-dimensional case, and we consider te solution of (13) wit a square of side lengt π. For tis domain, te positive eigenvalues can be computed by separation of variables. Tey are of te form m 2 + n 2 wit m and n integers: 1, 1, 2, 4, 4, 5, 5, 8,... If we approximate (13) using te space of continuous piecewise linear vector fields as te trial subspace of H(curl; ), te approximation fails badly. Tis is sown for an unstructured mes in Figure 2.5 and for a structured crisscross mes in Figure 2.6, were te nonzero discrete eigenvalues are plotted. Note te very different mode of failure for te two mes types. For more discussion of te spurious eigenvalues arising using continuous piecewise linear vector fields on a crisscross mes, see [20]. By contrast, if we use te lowest-order Raviart Tomas approximation of H(curl; ), as sown on te rigt of Figure 2.5, we obtain a provably good approximation for any mes. Tis is a very simple case of te general eigenvalue approximation teory presented in Section 3.6 below Figure 2.5. Approximation of te nonzero eigenvalues of (13) on an unstructured mes of te square (left) using continuous piecewise linear finite elements (middle) and Raviart Tomas elements (rigt). For te former, te discrete spectrum looks noting like te true spectrum, wile for te later it is very accurate.

Preconditioning in H(div) and Applications

Preconditioning in H(div) and Applications 1 Preconditioning in H(div) and Applications Douglas N. Arnold 1, Ricard S. Falk 2 and Ragnar Winter 3 4 Abstract. Summarizing te work of [AFW97], we sow ow to construct preconditioners using domain decomposition

More information

Smoothed projections in finite element exterior calculus

Smoothed projections in finite element exterior calculus Smooted projections in finite element exterior calculus Ragnar Winter CMA, University of Oslo Norway based on joint work wit: Douglas N. Arnold, Minnesota, Ricard S. Falk, Rutgers, and Snorre H. Cristiansen,

More information

FINITE ELEMENT EXTERIOR CALCULUS FOR PARABOLIC EVOLUTION PROBLEMS ON RIEMANNIAN HYPERSURFACES

FINITE ELEMENT EXTERIOR CALCULUS FOR PARABOLIC EVOLUTION PROBLEMS ON RIEMANNIAN HYPERSURFACES 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

More information

Finite Element Methods for Linear Elasticity

Finite Element Methods for Linear Elasticity Finite Element Metods for Linear Elasticity Ricard S. Falk Department of Matematics - Hill Center Rutgers, Te State University of New Jersey 110 Frelinguysen Rd., Piscataway, NJ 08854-8019 falk@mat.rutgers.edu

More information

ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL

ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 10, October 2014, Pages 5487 5502 S 0002-9947(2014)06134-5 Article electronically publised on February 26, 2014 ON THE CONSISTENCY OF

More information

arxiv: v1 [math.na] 20 Jul 2009

arxiv: v1 [math.na] 20 Jul 2009 STABILITY OF LAGRANGE ELEMENTS FOR THE MIXED LAPLACIAN DOUGLAS N. ARNOLD AND MARIE E. ROGNES arxiv:0907.3438v1 [mat.na] 20 Jul 2009 Abstract. Te stability properties of simple element coices for te mixed

More information

lecture 26: Richardson extrapolation

lecture 26: Richardson extrapolation 43 lecture 26: Ricardson extrapolation 35 Ricardson extrapolation, Romberg integration Trougout numerical analysis, one encounters procedures tat apply some simple approximation (eg, linear interpolation)

More information

Polynomial Interpolation

Polynomial Interpolation Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc

More information

Analysis of A Continuous Finite Element Method for H(curl, div)-elliptic Interface Problem

Analysis of A Continuous Finite Element Method for H(curl, div)-elliptic Interface Problem Analysis of A Continuous inite Element Metod for Hcurl, div)-elliptic Interface Problem Huoyuan Duan, Ping Lin, and Roger C. E. Tan Abstract In tis paper, we develop a continuous finite element metod for

More information

Polynomial Interpolation

Polynomial Interpolation Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x

More information

MIXED DISCONTINUOUS GALERKIN APPROXIMATION OF THE MAXWELL OPERATOR. SIAM J. Numer. Anal., Vol. 42 (2004), pp

MIXED DISCONTINUOUS GALERKIN APPROXIMATION OF THE MAXWELL OPERATOR. SIAM J. Numer. Anal., Vol. 42 (2004), pp MIXED DISCONTINUOUS GALERIN APPROXIMATION OF THE MAXWELL OPERATOR PAUL HOUSTON, ILARIA PERUGIA, AND DOMINI SCHÖTZAU SIAM J. Numer. Anal., Vol. 4 (004), pp. 434 459 Abstract. We introduce and analyze a

More information

1. Introduction. We consider the model problem: seeking an unknown function u satisfying

1. Introduction. We consider the model problem: seeking an unknown function u satisfying A DISCONTINUOUS LEAST-SQUARES FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS XIU YE AND SHANGYOU ZHANG Abstract In tis paper, a discontinuous least-squares (DLS) finite element metod is introduced

More information

A Hybrid Mixed Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems

A Hybrid Mixed Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems A Hybrid Mixed Discontinuous Galerkin Finite Element Metod for Convection-Diffusion Problems Herbert Egger Joacim Scöberl We propose and analyse a new finite element metod for convection diffusion problems

More information

LEAST-SQUARES FINITE ELEMENT APPROXIMATIONS TO SOLUTIONS OF INTERFACE PROBLEMS

LEAST-SQUARES FINITE ELEMENT APPROXIMATIONS TO SOLUTIONS OF INTERFACE PROBLEMS SIAM J. NUMER. ANAL. c 998 Society for Industrial Applied Matematics Vol. 35, No., pp. 393 405, February 998 00 LEAST-SQUARES FINITE ELEMENT APPROXIMATIONS TO SOLUTIONS OF INTERFACE PROBLEMS YANZHAO CAO

More information

The Laplace equation, cylindrically or spherically symmetric case

The Laplace equation, cylindrically or spherically symmetric case Numerisce Metoden II, 7 4, und Übungen, 7 5 Course Notes, Summer Term 7 Some material and exercises Te Laplace equation, cylindrically or sperically symmetric case Electric and gravitational potential,

More information

MATH745 Fall MATH745 Fall

MATH745 Fall MATH745 Fall MATH745 Fall 5 MATH745 Fall 5 INTRODUCTION WELCOME TO MATH 745 TOPICS IN NUMERICAL ANALYSIS Instructor: Dr Bartosz Protas Department of Matematics & Statistics Email: bprotas@mcmasterca Office HH 36, Ext

More information

How to Find the Derivative of a Function: Calculus 1

How to Find the Derivative of a Function: Calculus 1 Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te

More information

The derivative function

The derivative function Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative

More information

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx. Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions

More information

SMAI-JCM SMAI Journal of Computational Mathematics

SMAI-JCM SMAI Journal of Computational Mathematics SMAI-JCM SMAI Journal of Computational Matematics Compatible Maxwell solvers wit particles II: conforming and non-conforming 2D scemes wit a strong Faraday law Martin Campos Pinto & Eric Sonnendrücker

More information

A UNIFORM INF SUP CONDITION WITH APPLICATIONS TO PRECONDITIONING

A UNIFORM INF SUP CONDITION WITH APPLICATIONS TO PRECONDITIONING A UNIFORM INF SUP CONDIION WIH APPLICAIONS O PRECONDIIONING KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER Abstract. A uniform inf sup condition related to a parameter dependent Stokes problem is

More information

Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems

Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems Applied Matematics, 06, 7, 74-8 ttp://wwwscirporg/journal/am ISSN Online: 5-7393 ISSN Print: 5-7385 Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for

More information

SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY

SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY (Section 3.2: Derivative Functions and Differentiability) 3.2.1 SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY LEARNING OBJECTIVES Know, understand, and apply te Limit Definition of te Derivative

More information

Variational Localizations of the Dual Weighted Residual Estimator

Variational Localizations of the Dual Weighted Residual Estimator Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 Variational Localizations of te Dual Weigted Residual Estimator Tomas Ricter Tomas Wick Te dual weigted residual metod (DWR)

More information

Copyright c 2008 Kevin Long

Copyright c 2008 Kevin Long Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula

More information

A Mixed-Hybrid-Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems

A Mixed-Hybrid-Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems A Mixed-Hybrid-Discontinuous Galerkin Finite Element Metod for Convection-Diffusion Problems Herbert Egger Joacim Scöberl We propose and analyse a new finite element metod for convection diffusion problems

More information

New Streamfunction Approach for Magnetohydrodynamics

New Streamfunction Approach for Magnetohydrodynamics New Streamfunction Approac for Magnetoydrodynamics Kab Seo Kang Brooaven National Laboratory, Computational Science Center, Building 63, Room, Upton NY 973, USA. sang@bnl.gov Summary. We apply te finite

More information

Poisson Equation in Sobolev Spaces

Poisson Equation in Sobolev Spaces Poisson Equation in Sobolev Spaces OcMountain Dayligt Time. 6, 011 Today we discuss te Poisson equation in Sobolev spaces. It s existence, uniqueness, and regularity. Weak Solution. u = f in, u = g on

More information

CS522 - Partial Di erential Equations

CS522 - Partial Di erential Equations CS5 - Partial Di erential Equations Tibor Jánosi April 5, 5 Numerical Di erentiation In principle, di erentiation is a simple operation. Indeed, given a function speci ed as a closed-form formula, its

More information

APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS

APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS MATHEMATICS OF COMPUTATION Volume 71, Number 239, Pages 909 922 S 0025-5718(02)01439-4 Article electronically publised on Marc 22, 2002 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS DOUGLAS N. ARNOLD,

More information

5 Ordinary Differential Equations: Finite Difference Methods for Boundary Problems

5 Ordinary Differential Equations: Finite Difference Methods for Boundary Problems 5 Ordinary Differential Equations: Finite Difference Metods for Boundary Problems Read sections 10.1, 10.2, 10.4 Review questions 10.1 10.4, 10.8 10.9, 10.13 5.1 Introduction In te previous capters we

More information

arxiv: v1 [math.na] 17 Jul 2014

arxiv: v1 [math.na] 17 Jul 2014 Div First-Order System LL* FOSLL* for Second-Order Elliptic Partial Differential Equations Ziqiang Cai Rob Falgout Sun Zang arxiv:1407.4558v1 [mat.na] 17 Jul 2014 February 13, 2018 Abstract. Te first-order

More information

Differentiation in higher dimensions

Differentiation in higher dimensions Capter 2 Differentiation in iger dimensions 2.1 Te Total Derivative Recall tat if f : R R is a 1-variable function, and a R, we say tat f is differentiable at x = a if and only if te ratio f(a+) f(a) tends

More information

AN ANALYSIS OF NEW FINITE ELEMENT SPACES FOR MAXWELL S EQUATIONS

AN ANALYSIS OF NEW FINITE ELEMENT SPACES FOR MAXWELL S EQUATIONS Journal of Matematical Sciences: Advances and Applications Volume 5 8 Pages -9 Available at ttp://scientificadvances.co.in DOI: ttp://d.doi.org/.864/jmsaa_7975 AN ANALYSIS OF NEW FINITE ELEMENT SPACES

More information

ERROR BOUNDS FOR THE METHODS OF GLIMM, GODUNOV AND LEVEQUE BRADLEY J. LUCIER*

ERROR BOUNDS FOR THE METHODS OF GLIMM, GODUNOV AND LEVEQUE BRADLEY J. LUCIER* EO BOUNDS FO THE METHODS OF GLIMM, GODUNOV AND LEVEQUE BADLEY J. LUCIE* Abstract. Te expected error in L ) attimet for Glimm s sceme wen applied to a scalar conservation law is bounded by + 2 ) ) /2 T

More information

MA455 Manifolds Solutions 1 May 2008

MA455 Manifolds Solutions 1 May 2008 MA455 Manifolds Solutions 1 May 2008 1. (i) Given real numbers a < b, find a diffeomorpism (a, b) R. Solution: For example first map (a, b) to (0, π/2) and ten map (0, π/2) diffeomorpically to R using

More information

Order of Accuracy. ũ h u Ch p, (1)

Order of Accuracy. ũ h u Ch p, (1) Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical

More information

1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x)

1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x) Calculus. Gradients and te Derivative Q f(x+) δy P T δx R f(x) 0 x x+ Let P (x, f(x)) and Q(x+, f(x+)) denote two points on te curve of te function y = f(x) and let R denote te point of intersection of

More information

ch (for some fixed positive number c) reaching c

ch (for some fixed positive number c) reaching c GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 DOI 0.60/s4086-05-000-z Nonlinear Piecewise-defined Difference Equations wit Reciprocal and Cubic Terms Ramadan

More information

Stability, Consistency, and Convergence: A 21st Century Viewpoint. Convergence, consistency, and stability of discretizations

Stability, Consistency, and Convergence: A 21st Century Viewpoint. Convergence, consistency, and stability of discretizations Stability, Consistency, and Convergence: A 1st Century Viepoint Douglas N. Arnold Scool of Matematics, University of Minnesota Society for Industrial and Applied Matematics Feng Kang Distinguised Lecture

More information

Volume 29, Issue 3. Existence of competitive equilibrium in economies with multi-member households

Volume 29, Issue 3. Existence of competitive equilibrium in economies with multi-member households Volume 29, Issue 3 Existence of competitive equilibrium in economies wit multi-member ouseolds Noriisa Sato Graduate Scool of Economics, Waseda University Abstract Tis paper focuses on te existence of

More information

Symmetry Labeling of Molecular Energies

Symmetry Labeling of Molecular Energies Capter 7. Symmetry Labeling of Molecular Energies Notes: Most of te material presented in tis capter is taken from Bunker and Jensen 1998, Cap. 6, and Bunker and Jensen 2005, Cap. 7. 7.1 Hamiltonian Symmetry

More information

Chapter 5 FINITE DIFFERENCE METHOD (FDM)

Chapter 5 FINITE DIFFERENCE METHOD (FDM) MEE7 Computer Modeling Tecniques in Engineering Capter 5 FINITE DIFFERENCE METHOD (FDM) 5. Introduction to FDM Te finite difference tecniques are based upon approximations wic permit replacing differential

More information

A Reconsideration of Matter Waves

A Reconsideration of Matter Waves A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,

More information

Discontinuous Galerkin Methods for Relativistic Vlasov-Maxwell System

Discontinuous Galerkin Methods for Relativistic Vlasov-Maxwell System Discontinuous Galerkin Metods for Relativistic Vlasov-Maxwell System He Yang and Fengyan Li December 1, 16 Abstract e relativistic Vlasov-Maxwell (RVM) system is a kinetic model tat describes te dynamics

More information

Robotic manipulation project

Robotic manipulation project Robotic manipulation project Bin Nguyen December 5, 2006 Abstract Tis is te draft report for Robotic Manipulation s class project. Te cosen project aims to understand and implement Kevin Egan s non-convex

More information

Inf sup testing of upwind methods

Inf sup testing of upwind methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Met. Engng 000; 48:745 760 Inf sup testing of upwind metods Klaus-Jurgen Bate 1; ;, Dena Hendriana 1, Franco Brezzi and Giancarlo

More information

Derivatives. By: OpenStaxCollege

Derivatives. By: OpenStaxCollege By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator

More information

arxiv: v1 [math.na] 28 Apr 2017

arxiv: v1 [math.na] 28 Apr 2017 THE SCOTT-VOGELIUS FINITE ELEMENTS REVISITED JOHNNY GUZMÁN AND L RIDGWAY SCOTT arxiv:170500020v1 [matna] 28 Apr 2017 Abstract We prove tat te Scott-Vogelius finite elements are inf-sup stable on sape-regular

More information

Mass Lumping for Constant Density Acoustics

Mass Lumping for Constant Density Acoustics Lumping 1 Mass Lumping for Constant Density Acoustics William W. Symes ABSTRACT Mass lumping provides an avenue for efficient time-stepping of time-dependent problems wit conforming finite element spatial

More information

A SYMMETRIC NODAL CONSERVATIVE FINITE ELEMENT METHOD FOR THE DARCY EQUATION

A SYMMETRIC NODAL CONSERVATIVE FINITE ELEMENT METHOD FOR THE DARCY EQUATION A SYMMETRIC NODAL CONSERVATIVE FINITE ELEMENT METHOD FOR THE DARCY EQUATION GABRIEL R. BARRENECHEA, LEOPOLDO P. FRANCA 1 2, AND FRÉDÉRIC VALENTIN Abstract. Tis work introduces and analyzes novel stable

More information

MANY scientific and engineering problems can be

MANY scientific and engineering problems can be A Domain Decomposition Metod using Elliptical Arc Artificial Boundary for Exterior Problems Yajun Cen, and Qikui Du Abstract In tis paper, a Diriclet-Neumann alternating metod using elliptical arc artificial

More information

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example, NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing

More information

Finite Difference Methods Assignments

Finite Difference Methods Assignments Finite Difference Metods Assignments Anders Söberg and Aay Saxena, Micael Tuné, and Maria Westermarck Revised: Jarmo Rantakokko June 6, 1999 Teknisk databeandling Assignment 1: A one-dimensional eat equation

More information

LECTURE 14 NUMERICAL INTEGRATION. Find

LECTURE 14 NUMERICAL INTEGRATION. Find LECTURE 14 NUMERCAL NTEGRATON Find b a fxdx or b a vx ux fx ydy dx Often integration is required. However te form of fx may be suc tat analytical integration would be very difficult or impossible. Use

More information

Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 01 Aug 08, 2016.

Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 01 Aug 08, 2016. Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals Gary D. Simpson gsim1887@aol.com rev 1 Aug 8, 216 Summary Definitions are presented for "quaternion functions" of a quaternion. Polynomial

More information

Lecture 15. Interpolation II. 2 Piecewise polynomial interpolation Hermite splines

Lecture 15. Interpolation II. 2 Piecewise polynomial interpolation Hermite splines Lecture 5 Interpolation II Introduction In te previous lecture we focused primarily on polynomial interpolation of a set of n points. A difficulty we observed is tat wen n is large, our polynomial as to

More information

A Finite Element Primer

A Finite Element Primer A Finite Element Primer David J. Silvester Scool of Matematics, University of Mancester d.silvester@mancester.ac.uk. Version.3 updated 4 October Contents A Model Diffusion Problem.................... x.

More information

arxiv: v2 [math.na] 5 Jul 2017

arxiv: v2 [math.na] 5 Jul 2017 Trace Finite Element Metods for PDEs on Surfaces Maxim A. Olsanskii and Arnold Reusken arxiv:1612.00054v2 [mat.na] 5 Jul 2017 Abstract In tis paper we consider a class of unfitted finite element metods

More information

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note

More information

2.8 The Derivative as a Function

2.8 The Derivative as a Function .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open

More information

Introduction to Derivatives

Introduction to Derivatives Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))

More information

Exam 1 Review Solutions

Exam 1 Review Solutions Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),

More information

Teaching Differentiation: A Rare Case for the Problem of the Slope of the Tangent Line

Teaching Differentiation: A Rare Case for the Problem of the Slope of the Tangent Line Teacing Differentiation: A Rare Case for te Problem of te Slope of te Tangent Line arxiv:1805.00343v1 [mat.ho] 29 Apr 2018 Roman Kvasov Department of Matematics University of Puerto Rico at Aguadilla Aguadilla,

More information

Computers and Mathematics with Applications. A nonlinear weighted least-squares finite element method for Stokes equations

Computers and Mathematics with Applications. A nonlinear weighted least-squares finite element method for Stokes equations Computers Matematics wit Applications 59 () 5 4 Contents lists available at ScienceDirect Computers Matematics wit Applications journal omepage: www.elsevier.com/locate/camwa A nonlinear weigted least-squares

More information

Parameter Fitted Scheme for Singularly Perturbed Delay Differential Equations

Parameter Fitted Scheme for Singularly Perturbed Delay Differential Equations International Journal of Applied Science and Engineering 2013. 11, 4: 361-373 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Awoke Andargiea* and Y. N. Reddyb a b Department

More information

Jian-Guo Liu 1 and Chi-Wang Shu 2

Jian-Guo Liu 1 and Chi-Wang Shu 2 Journal of Computational Pysics 60, 577 596 (000) doi:0.006/jcp.000.6475, available online at ttp://www.idealibrary.com on Jian-Guo Liu and Ci-Wang Su Institute for Pysical Science and Tecnology and Department

More information

On convergence of the immersed boundary method for elliptic interface problems

On convergence of the immersed boundary method for elliptic interface problems On convergence of te immersed boundary metod for elliptic interface problems Zilin Li January 26, 2012 Abstract Peskin s Immersed Boundary (IB) metod is one of te most popular numerical metods for many

More information

A = h w (1) Error Analysis Physics 141

A = h w (1) Error Analysis Physics 141 Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.

More information

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225 THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:

More information

Cubic Functions: Local Analysis

Cubic Functions: Local Analysis Cubic function cubing coefficient Capter 13 Cubic Functions: Local Analysis Input-Output Pairs, 378 Normalized Input-Output Rule, 380 Local I-O Rule Near, 382 Local Grap Near, 384 Types of Local Graps

More information

WYSE Academic Challenge 2004 Sectional Mathematics Solution Set

WYSE Academic Challenge 2004 Sectional Mathematics Solution Set WYSE Academic Callenge 00 Sectional Matematics Solution Set. Answer: B. Since te equation can be written in te form x + y, we ave a major 5 semi-axis of lengt 5 and minor semi-axis of lengt. Tis means

More information

Key words. Sixth order problem, higher order partial differential equations, biharmonic problem, mixed finite elements, error estimates.

Key words. Sixth order problem, higher order partial differential equations, biharmonic problem, mixed finite elements, error estimates. A MIXED FINITE ELEMENT METHOD FOR A SIXTH ORDER ELLIPTIC PROBLEM JÉRÔME DRONIOU, MUHAMMAD ILYAS, BISHNU P. LAMICHHANE, AND GLEN E. WHEELER Abstract. We consider a saddle point formulation for a sixt order

More information

FINITE ELEMENT APPROXIMATIONS AND THE DIRICHLET PROBLEM FOR SURFACES OF PRESCRIBED MEAN CURVATURE

FINITE ELEMENT APPROXIMATIONS AND THE DIRICHLET PROBLEM FOR SURFACES OF PRESCRIBED MEAN CURVATURE FINITE ELEMENT APPROXIMATIONS AND THE DIRICHLET PROBLEM FOR SURFACES OF PRESCRIBED MEAN CURVATURE GERHARD DZIUK AND JOHN E. HUTCHINSON Abstract. We give a finite element procedure for te Diriclet Problem

More information

Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Journal of Computational and Applied Matematics 94 (6) 75 96 Contents lists available at ScienceDirect Journal of Computational and Applied Matematics journal omepage: www.elsevier.com/locate/cam Smootness-Increasing

More information

c 2004 Society for Industrial and Applied Mathematics

c 2004 Society for Industrial and Applied Mathematics SIAM J NUMER ANAL Vol 4, No, pp 86 84 c 004 Society for Industrial and Applied Matematics LEAST-SQUARES METHODS FOR LINEAR ELASTICITY ZHIQIANG CAI AND GERHARD STARKE Abstract Tis paper develops least-squares

More information

5.1 We will begin this section with the definition of a rational expression. We

5.1 We will begin this section with the definition of a rational expression. We Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go

More information

Click here to see an animation of the derivative

Click here to see an animation of the derivative Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,

More information

A SADDLE POINT LEAST SQUARES APPROACH TO MIXED METHODS

A SADDLE POINT LEAST SQUARES APPROACH TO MIXED METHODS A SADDLE POINT LEAST SQUARES APPROACH TO MIXED METHODS CONSTANTIN BACUTA AND KLAJDI QIRKO Abstract. We investigate new PDE discretization approaces for solving variational formulations wit different types

More information

A MONTE CARLO ANALYSIS OF THE EFFECTS OF COVARIANCE ON PROPAGATED UNCERTAINTIES

A MONTE CARLO ANALYSIS OF THE EFFECTS OF COVARIANCE ON PROPAGATED UNCERTAINTIES A MONTE CARLO ANALYSIS OF THE EFFECTS OF COVARIANCE ON PROPAGATED UNCERTAINTIES Ronald Ainswort Hart Scientific, American Fork UT, USA ABSTRACT Reports of calibration typically provide total combined uncertainties

More information

A First-Order System Approach for Diffusion Equation. I. Second-Order Residual-Distribution Schemes

A First-Order System Approach for Diffusion Equation. I. Second-Order Residual-Distribution Schemes A First-Order System Approac for Diffusion Equation. I. Second-Order Residual-Distribution Scemes Hiroaki Nisikawa W. M. Keck Foundation Laboratory for Computational Fluid Dynamics, Department of Aerospace

More information

Different Approaches to a Posteriori Error Analysis of the Discontinuous Galerkin Method

Different Approaches to a Posteriori Error Analysis of the Discontinuous Galerkin Method WDS'10 Proceedings of Contributed Papers, Part I, 151 156, 2010. ISBN 978-80-7378-139-2 MATFYZPRESS Different Approaces to a Posteriori Error Analysis of te Discontinuous Galerkin Metod I. Šebestová Carles

More information

Lecture 21. Numerical differentiation. f ( x+h) f ( x) h h

Lecture 21. Numerical differentiation. f ( x+h) f ( x) h h Lecture Numerical differentiation Introduction We can analytically calculate te derivative of any elementary function, so tere migt seem to be no motivation for calculating derivatives numerically. However

More information

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x

More information

Numerical Differentiation

Numerical Differentiation Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function

More information

Financial Econometrics Prof. Massimo Guidolin

Financial Econometrics Prof. Massimo Guidolin CLEFIN A.A. 2010/2011 Financial Econometrics Prof. Massimo Guidolin A Quick Review of Basic Estimation Metods 1. Were te OLS World Ends... Consider two time series 1: = { 1 2 } and 1: = { 1 2 }. At tis

More information

Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations

Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations Arbitrary order exactly divergence-free central discontinuous Galerkin metods for ideal MHD equations Fengyan Li, Liwei Xu Department of Matematical Sciences, Rensselaer Polytecnic Institute, Troy, NY

More information

HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS. PART I: THE STOKES PROBLEM

HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS. PART I: THE STOKES PROBLEM MATHEMATICS OF COMPUTATION Volume 75, Number 254, Pages 533 563 S 0025-5718(05)01804-1 Article electronically publised on December 16, 2005 HYBRIDIZED GLOBALLY DIVERGENCE-FREE LDG METHODS. PART I: THE

More information

Combining functions: algebraic methods

Combining functions: algebraic methods Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

More information

A SHORT INTRODUCTION TO BANACH LATTICES AND

A SHORT INTRODUCTION TO BANACH LATTICES AND CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,

More information

Reflection Symmetries of q-bernoulli Polynomials

Reflection Symmetries of q-bernoulli Polynomials Journal of Nonlinear Matematical Pysics Volume 1, Supplement 1 005, 41 4 Birtday Issue Reflection Symmetries of q-bernoulli Polynomials Boris A KUPERSHMIDT Te University of Tennessee Space Institute Tullaoma,

More information

Higher Derivatives. Differentiable Functions

Higher Derivatives. Differentiable Functions Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.

More information

Downloaded 11/15/17 to Redistribution subject to SIAM license or copyright; see

Downloaded 11/15/17 to Redistribution subject to SIAM license or copyright; see SIAM J. NUMER. ANAL. Vol. 55, No. 6, pp. 2787 2810 c 2017 Society for Industrial and Applied Matematics EDGE ELEMENT METHOD FOR OPTIMAL CONTROL OF STATIONARY MAXWELL SYSTEM WITH GAUSS LAW IRWIN YOUSEPT

More information

arxiv: v1 [math.na] 19 Mar 2018

arxiv: v1 [math.na] 19 Mar 2018 A primal discontinuous Galerkin metod wit static condensation on very general meses arxiv:1803.06846v1 [mat.na] 19 Mar 018 Alexei Lozinski Laboratoire de Matématiques de Besançon, CNRS UMR 663, Univ. Bourgogne

More information

arxiv: v1 [math.na] 27 Jan 2014

arxiv: v1 [math.na] 27 Jan 2014 L 2 -ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF BOUNDARY FLUXES MATS G. LARSON AND ANDRÉ MASSING arxiv:1401.6994v1 [mat.na] 27 Jan 2014 Abstract. We prove quasi-optimal a priori error estimates

More information

Function Composition and Chain Rules

Function Composition and Chain Rules Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function

More information

Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser

More information

AN ANALYSIS OF THE EMBEDDED DISCONTINUOUS GALERKIN METHOD FOR SECOND ORDER ELLIPTIC PROBLEMS

AN ANALYSIS OF THE EMBEDDED DISCONTINUOUS GALERKIN METHOD FOR SECOND ORDER ELLIPTIC PROBLEMS AN ANALYSIS OF THE EMBEDDED DISCONTINUOUS GALERKIN METHOD FOR SECOND ORDER ELLIPTIC PROBLEMS BERNARDO COCKBURN, JOHNNY GUZMÁN, SEE-CHEW SOON, AND HENRYK K. STOLARSKI Abstract. Te embedded discontinuous

More information

CHAPTER 4. Elliptic PDEs

CHAPTER 4. Elliptic PDEs CHAPTER 4 Elliptic PDEs One of te main advantages of extending te class of solutions of a PDE from classical solutions wit continuous derivatives to weak solutions wit weak derivatives is tat it is easier

More information

arxiv: v1 [math.na] 9 Sep 2015

arxiv: v1 [math.na] 9 Sep 2015 arxiv:509.02595v [mat.na] 9 Sep 205 An Expandable Local and Parallel Two-Grid Finite Element Sceme Yanren ou, GuangZi Du Abstract An expandable local and parallel two-grid finite element sceme based on

More information