Adaptive Finite Element Algorithms

Transcription

1 Adaptive Finite Element Algorithms of optimal complexity Adaptieve Eindige Elementen Algoritmen van optimale complexiteit (met een samenvatting in het Nederlands) PROEFSCHRIF ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus prof. dr. W.H. Gispen, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 18 oktober 2006 des middags te uur door YAROSLAV KONDRAYUK geboren op 12 juli 1977 te Khmelnitsk reg., Oekraine

2 Promotor: Co-promotor: Prof. dr. H.A. van der Vorst Dr. R.P. Stevenson Het onderzoek in dit proefschrift is financieel mogelijk gemaakt door de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) Mathematics Subject Classification: 41A25, 65N12, 65N30, 65N50, 65Y20, 76M10 Yaroslav Kondratyuk Adaptive finite element algorithms of optimal complexity Proefschrift Universiteit Utrecht Met een samenvatting in het Nederlands ISBN

3 Contents 1 Introduction Scope of problems and motivations Navier-Stokes equations Linearized and stationary Navier-Stokes problem of Oseen type Stokes Problem hesis overview Chapter Chapter Chapter Chapter Analysis of Adaptive FE Algorithms for the Poisson Problem Abstract Variational Problem Galerkin Method Finite Element Approximation Spaces Local Mesh Refinement, Adaptivity and Optimal Complexity Algorithms Linear versus nonlinear and adaptive approximations Singular solutions of boundary value problems Nonconforming triangulations and hierarchical tree-structures Adaptive Algorithms of Optimal Complexity A Posteriori Error Analysis Basic Principle of Adaptive Error Reduction First Practical Convergent Adaptive FE Algorithm Second Practical Adaptive FE Algorithm Optimal Adaptive Algorithms Multiscale Decomposition of FEM Approximation Spaces Optimization of Adaptive riangulations Optimal Adaptive Finite Element Algorithm Numerical Experiments L-shaped domain Domain with a crack i

4 3 Adaptive FE Algorithms for Singularly Perturbed Problems Singularly Perturbed Reaction-Diffusion Equation Robust Residual Based A Posteriori Error Estimator Uniformly Convergent Adaptive Algorithm Numerical Example Derefinement-based Adaptive FE Algorithms for the Stokes Problem Introduction Mixed Variational Problems. Well-Posedness and Discretization Stokes Problem Methods of Optimal Complexity for Solving the Stokes Problem Derefinement/optimization of finite element approximation Derefinement of pressure approximation Derefinement of velocity approximation Adaptive Fixed Point Algorithms for Stokes Problem: Analysis and Convergence Convergent Adaptive Finite Element Algorithm for Inner Elliptic Problem Conclusions An Optimal Adaptive FEM for the Stokes Problem Introduction Stokes problem Finite element approximation Overview of the solution method A posteriori error estimators A posteriori error estimator for the inner elliptic problem A posteriori error estimator for the (reduced) Stokes problem Adaptive refinements resulting in error reduction Adaptive pressure refinements Adaptive velocity refinements An adaptive method for the Stokes problem in an idealized setting A practical adaptive method for the Stokes problem Numerical experiment Nederlandse samenvatting 139 Dankwoord/Acknowledgements 142 Curriculum Vitae 144

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6 Chapter 1 Introduction his thesis is dedicated to the design and analysis of optimal algorithms for solving partial differential equations (PDEs). Recent tremendous progress in the performance of modern computers stimulated the development of efficient numerical methods. Modern numerical simulations employing large-scale computations provide a powerful approach for real-life problems. For the development of new numerical models, verification by means of rigorous mathematical analysis and numerical computations has become essential. he explanation by theoretical analysis, detailed analysis based on numerical computations, and also the comparison of these with real experiments, altogether usually guide the development of numerical models. Nevertheless, the most difficult problems, for instance turbulence in fluid flow, remain to be challenging problems for modern computers and numerical methods. In view of this, the design of optimal algorithms has received special attention in the numerical community. In this thesis, we are primarily interested in the development of efficient algorithms, with respect to computer memory usage and execution time, for solving numerically PDEs. Aiming for an optimal algorithm, we are naturally led to the so-called adaptive method. Simply speaking, the task of an adaptive method is to adapt the approximation to the unknown solution of a differential equation during the solution process, using only a posteriori information, and to produce eventually the (quasi-) best possible approximation with optimal computational costs. Nowadays adaptive finite element algorithms are being used to solve efficiently partial differential equations (PDEs) arising in science and engineering [40, 1, 3]. Adaptive methods are particularly efficient if the solution of a boundary value problem has singularities. he origin of a singularity can be a concentration of stresses in elasticity, boundary/internal layers in environmental pollution problems, or a propagation of shock-waves in fluid mechanics. he general structure of the loop of an adaptive algorithm is Solve Estimate Refine, Derefine. A fully adaptive finite element algorithm solves a discrete problem on some triangulation, estimates the error in the current approximation using an a posteriori error estimator and performs adaptive refinement of the triangulation in the regions where the error is largest. Using such an adaptive procedure, we work with a sequence of successively refined 1

7 triangulations with the objective to satisfy a stopping criterion based on the a posteriori error estimate with minimal computational effort. Despite the big success of adaptive FE methods in practical applications during the last 25 years, they were not even proven to converge in more than one space dimension before the works of Dörfler [19] (in 1996), and Morin, Nochetto, Siebert [29] (in 2000). Convergence alone, however, does not imply that the adaptive method is more efficient than its non-adaptive counterpart. Using the results concerning convergence, afterwards Binev, Dahmen and DeVore [6] (in 2004), and Stevenson [36] (in 2005) showed that adaptive FEMs converge with optimal rates and with linear computational complexity. hese first optimality results were based on the use of special mesh coarsening routines. In those references and in this thesis, an adaptive finite element method is called optimal if it performs as follows: whenever for some s > 0, the solution of boundary value problem can be approximated within a tolerance ε > 0, in energy norm, by a finite element approximation on some triangulation with O(ε 1/s ) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate, then the adaptive finite element method produces approximations that converge with this rate, with a number of operations that is of the order of the number of triangles in the output triangulation. 1.1 Scope of problems and motivations Navier-Stokes equations he Navier-Stokes problem (1.1.1) is a very difficult problem to solve numerically. u ν u + (u )u + p = f in Ω (0, ) t divu = 0 in Ω u = 0 on Ω (0, ) (1.1.1) ypically, the solutions of this problem show strong singularities, and the application of adaptive procedures is very valuable in this case. In view of the nonstationary behavior of singularities, a detailed analysis of optimality of local refinement/derefinement is necessary Linearized and stationary Navier-Stokes problem of Oseen type ypically, an Oseen problem (1.1.2) has to be solved at each step in the numerical approximation of the Navier-Stokes problem. ν u + (b )u + cu + p = f in Ω divu = 0 in Ω (1.1.2) u = 0 on Ω his problem too is computationally far from trivial, especially because of singular perturbations caused by the interactions between diffusion, convection and reaction terms. 2

8 o our knowledge, no complete analysis of optimal adaptive algorithms exists for this type of problem Stokes Problem he Stokes problem (1.1.3) is one of the main problems in this thesis, for which we study and develop optimal adaptive algorithms. u + p = f in Ω divu = 0 in Ω (1.1.3) u = 0 on Ω. 1.2 hesis overview Chapter 2 In this chapter we explain in detail the most important recent achievements in the theory of optimal adaptive FE algorithms, working with the Poisson problem. First, after recalling the basic properties of the finite element method, we consider the problem of approximation of a given function, and explain the notions of linear, nonlinear, and adaptive approximation. hen we explain why we may expect advantages of adaptive approximation for the numerical solution of partial differential equations. Furthermore, the construction and analysis of a posteriori error estimators is presented, and we discuss how to design a convergent adaptive method. We explain the notion of an optimal algorithm and present the construction of an algorithm that produces a sequence of approximations that converges with the optimal rate. Finally, we also present a proof for the optimal computational complexity of this adaptive algorithm. We wrote a C++ code implementing a 2D adaptive finite element solver. In this algorithm, Galerkin systems are solved with CG and a wavelet preconditioner. he solver has the possibility to switch between a nodal FEM basis and a wavelet basis on adaptive triangulations, when multiscale analysis of the solution is required. o demonstrate the performance of our solver, we present numerical examples for the Poisson problem on an L-shaped domain and on a domain with a crack Chapter 3 Here, we study adaptive algorithms for the singularly perturbed reaction-diffusion problem. Problems of this type arise, for example, in the modeling of thin plates, shells, in the numerical treatment of reaction processes, and as a result of an implicit time semidiscretization of parabolic or hyperbolic equations of second order. he main feature of the reaction-diffusion equation is that its solution exhibits very steep boundary layers for large values of the reaction term. In view of this, the application of adaptive methods to solve this problem is particularly attractive ([33, 42]). We are interested in adaptive methods that converge uniformly in terms of the reaction term. 3

9 he main ingredients of a robust solver for this kind of problem are discussed. In particular we present an analysis of a robust a posteriori error estimator and a uniform saturation property. We have implemented in C++ a 2D adaptive solver for the singularly perturbed reaction-diffusion problem. his solver constructs a sequence of adaptive approximations that converge uniformly in terms of the reaction term. Usually, it is quite difficult to achieve linear computational complexity for an adaptive algorithm in practice, because this requires a careful design and programming. We did numerical experiments that indicate that our adaptive solver is quite fast in operating with hundreds of thousands degrees of freedom and that it has linear computational complexity Chapter 4 his chapter and the paper [25] are dedicated to the design of optimal algorithms for the Stokes problem. ypically, problems in fluid mechanics (1.1.1), (1.1.2), (1.1.3) naturally lead to mixed variational problems [21, 11]. For the adaptive solution of mixed variational problems, the situation is much more complicated than for elliptic problems, and we are not aware of any proof of optimality of adaptive finite element algorithms. Bänch, Morin and Nochetto [5] introduced an adaptive FEM for the Stokes problem, and they proved convergence of the method. Since convergence alone does not imply that the adaptive method is more efficient than its non-adaptive counterpart, an analysis of the convergence rates is important. he numerical experiments in [5] show (quasi-) optimal triangulations for some values of the parameters, but a theoretical explanation of these results is still an open question. In [25] and in this chapter we present a detailed design of an adaptive FE algorithm for the Stokes problem, and analyze its computational complexity. As in [5], we apply a fixed point iteration to an infinite dimensional Schur complement operator. For the approximation of the inverse of the elliptic operator we use a generalization of the optimal adaptive finite element method from Chapter 2 for elliptic equations. By adding a derefinement step to the resulting adaptive fixed point iteration, in order to optimize the underlying triangulation after each fixed number of iterations, we show that the resulting method converges with the optimal rate and that it has optimal computational complexity. Apart from its use here, in our view, an efficient mesh optimization/derefinement routine could be very useful for the adaptive solution of nonstationary problems that exhibit moving shock waves Chapter 5 Recently, in the work of Stevenson [37], an adaptive FEM method was proposed for solving elliptic problems that has optimal computational complexity not relying on a recurrent derefinement of the triangulations. his result is very valuable for the construction of optimal adaptive algorithms for stationary problems. he extention of this result to the Stokes mixed variational problem appeared to be very complicated; it is the subject of this chapter (that presents a joint work with Stevenson [27]). 4

10 A new adaptive finite element method for solving the Stokes equations is developed, which is shown to converge with the best possible rate. he method consists of 3 nested loops. For a sequence of finite element spaces with respect to adaptively refined partitions, the solution is approximated in the outmost loop by that of the Stokes problem in which the divergence-free condition is reduced to orthogonality of the divergence to the finite element space in which the approximate pressure is sought. For solving each of these semi-discrete problems, the Uzawa iteration is applied. Finally, the elliptic system for the velocity that has to be solved in each Uzawa step is approximately solved by an adaptive finite element method. We have implemented this adaptive algorithm and we present a discussion of the numerical performance of our method for the Stokes problem on an L-shaped domain. We compare our results with those from an adaptive Uzawa method introduced by Bänch, Morin and Nochetto in [5]. 5

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12 Chapter 2 Analysis of Adaptive FE Algorithms for the Poisson Problem 2.1 Abstract Variational Problem Let W be a real Hilbert space endowed with norm. W, and let W denote its dual space, i.e., the space of all bounded linear functionals f : W R equipped with the norm f W := Consider the following abstract variational problem (AVP) f(w) sup. (2.1.1) w W,w 0 w W Given f W, find u W such that (2.1.2) a(u, w) = f(w) for all w W. Definition A bilinear form a(.,.) : W W R is called continuous if there exists a constant ς > 0 such that a(u, w) ς u W w W for all u, w W. (2.1.3) Definition A bilinear form a(.,.) is called W -elliptic (or coercive), if there exists a constant α > 0 such that, a(w, w) α w 2 W for all w W. (2.1.4) he question about existence and uniqueness of the solution of (AVP) is answered by the Lax-Milgram lemma ([35, 13]): 7

13 Lemma (Lax-Milgram) Let W be a (real) Hilbert space, a(.,.) : W W R a bilinear, continuous and W-elliptic form, and f : W R be a bounded linear functional. hen there exists a unique solution u W to (AVP), and moreover where the constant α stems from the coercivity condition. u W 1 α f W, (2.1.5) Galerkin Method In the following let us assume that the conditions of the Lax-Milgram lemma hold. With the aim to build a numerical approximation to the solution of the abstract variational problem, we choose a sequence of subspaces (W j ) j N W, such that for all w W inf w w j W 0 as j. (2.1.6) w j W j For each W j W we can formulate the following discrete problem Given f W, find u j W j such that (2.1.7) a(u j, w j ) = f(w j ) for all w j W j. Since W j is a subspace of W, the conditions of Lax-Milgram lemma are true also for the discrete problem providing the existence and uniqueness of the discrete solution u j, that is called a Galerkin approximation. More precisely, its properties are stated in the following theorem([32]). heorem (Properties of a Galerkin approximation) Under the assumptions of the Lax-Milgram lemma, there exists a unique solution u j to (2.1.7). Furthermore it is stable, since independently of the choice of the subspace W j it satisfies Moreover, with u being the solution of (AVP), we have that u j W 1 α f W. (2.1.8) u u j W ς α inf w j W j u w j W. (2.1.9) In the following we consider the special case that the bilinear form a(.,.) is symmetric. he assumptions concerning symmetry, continuity and coercivity imply that we can introduce an energy inner product and energy norm induced by a(.,.): (u, w) E := a(u, w) for all u, w W, (2.1.10) 8

14 and the following norm equivalence holds u E := a(u, u) 1/2 for all u W, (2.1.11) α 1/2 u W u E ς 1/2 u W for all u W. (2.1.12) Now, let u and u j be the solutions of (AVP) and its discrete analog (2.1.7), respectively. Comparing (2.1.2) and (2.1.7) we infer that a(u u j, w j ) = 0 for all w j W j (2.1.13) showing that the error u u j of the Galerkin approximation is orthogonal to the approximation space W j with respect to the energy inner product. As a consequence, we have u w j 2 E = u j w j 2 E + u u j 2 E for all w j W, (2.1.14) and so the following lemma clearly holds true. Lemma If, in addition to the conditions of the Lax-Milgam lemma, the bilinear form a(.,.) : W W R is symmetric, then the Galerkin approximation u j, being the solution of (2.1.7), is the best approximation to the solution of (AVP) from the approximation space W j, i.e., u u j E = inf u w j E. (2.1.15) w j W j So note that in view of estimate (2.1.9) and property (2.1.6), we have lim u j = u in W. (2.1.16) j However, this result says nothing about the rate of convergence of the approximate solutions towards the exact one. For this we need additional information, namely, properties of the approximation spaces W j and the regularity of the solution. One of the possible choices for W j is provided by the finite element method. 2.2 Finite Element Approximation Spaces As we have seen in the previous section, to use the Galerkin method for the numerical solution of (AVP) we need to specify the approximation spaces W j. he finite element method is a specific procedure of constructing computationally efficient approximation spaces which are called finite element spaces. In many cases of practical interest we deal with a boundary value problem posed over a domain Ω R d with Lipschitz-continuous boundary. Depending on the boundary value problem under consideration, W is for example one of the spaces H 1 0(Ω), H 1 (Ω), H 2 0(Ω), H(div, Ω), H(curl, Ω) etc. Here we shall confine ourselves mostly to variational problems defined on H 1 0(Ω). his corresponds to 9

15 a second order elliptic boundary value problem. In particular, we start with the Poisson problem a(u, w) := Ω Given f H 1 (Ω) := (H 1 0(Ω)), find u H 1 0(Ω) such that (2.2.1) u w = f(w) for all w H 1 0(Ω) where we assume that the domain Ω is a polytope. he bilinear form a(, ) : H 1 0(Ω) H 1 0(Ω) R is continuous, coercive and symmetric. So on H 1 0(Ω), the H 1 0 (Ω)-norm and the energy-norm E differ at most a constant multiple. We switch between these norms at our convenience. Defining A : H 1 0(Ω) H 1 (Ω) by (A u)(w) = a(u, w), (2.2.1) can be rewritten as A u = f. (2.2.2) We will measure the error of any approximation for u in the energy norm E, with dual norm f(w) f E := sup, (f H 1 (Ω)). (2.2.3) 0 w H0 1(Ω) w E Equipped with these norms, A : H 1 0(Ω) H 1 (Ω) is an isomorphism. he first step within the finite element procedure is to produce a triangulation of the domain Ω ([12]), i.e., a finite collection of closed polytopes K (the elements ) such that Ω = K K the interior of each polytope K is non empty, i.e., int(k) for each pair of distinct K 1, K 2 one has int(k 1 ) int(k 2 ) = In view of constructing a sequence of subspaces (W j ) j N we will consider a sequence of triangulations ( j ) j N. Definition A sequence of triangulations ( j ) j N is called quasi-uniform if max diam(k) min diam(k). (2.2.4) K j K j Here and throughout the thesis, in order to avoid the repeated use of generic but unspecified constants, by C D we mean that C can be bounded by a multiple of D, independently of parameters which C and D may depend on, where in particular we think of j. Obviously, C D is defined as D C, and by C D we mean that C D and C D. Definition If for any pair of distinct polytopes K 1, K 2 with K 1 K 2, their intersection is a common lower dimensional face of K 1 and K 2, i.e., for d = 3 it is either a common face, side or vertex, then the triangulation is called conforming and otherwise we shall call it a nonconforming triangulation. A vertex of any K is called a vertex of. It is called a non-hanging vertex if it is a vertex for all K that contain it, otherwise it is called a hanging vertex (see Fig. 2.1). 10

16 Figure 2.1: Non-hanging and hanging vertices. Once we set up the triangulation j, the next step is to construct the corresponding approximation space W j. For some k N >0, for each element K j let P (K) be a space of algebraic polynomials of some fixed finite dimension that includes the space P k (K) of all polynomials of total degree less or equal to k. We define W j := {v C r (Ω) K j P (K) : v Ω = 0} (2.2.5) where r { 1, 0, 1, } and C 1 (Ω) := L 2 (Ω). Depending on k and of the type of polytopes being used, the space P (K) will be chosen to be sufficiently large such that, despite of the intersection with C r (Ω), the following direct or Jackson estimate is valid: For all p [1, ], s t k + 1, s < r + 1 p inf u w j W s w j W p (Ω) h t s j u W t p (Ω) for all u W t j p(ω) (2.2.6) where h j := max K j diam(k). Since measured in L 2 (Ω), the error behaves at best like O(h k+1 ), the space W j is said to have order k + 1. Defining, for a triangulation, A : W (W ) H 1 (Ω) by (A v )(w ) = a(v, w ) for v, w W, u := A 1 f is the Galerkin approximation of u. One easily verifies that for any f H 1 (Ω), A 1 f E f E. Definition If, for each polytope K j, the ratio of the radii of the smallest circumscribed and the largest inscribed ball of K is bounded uniformly in K j and j N, then the sequence of triangulations ( j ) j N is called shape-regular. In finite element analysis we need the shape regularity requirement to avoid very small angles in the triangulation since typical FEM estimates depend on the minimal angle of the triangulation and deteriorate when it becomes small. For a shape regular sequence of triangulations, we have the following inverse or Bernstein estimate (e.g., [31]): With h j := min K j diam(k) and s t < r + 1 p, w j W t p (Ω) h s t j w j W s p (Ω) for all w j W j. (2.2.7) Since W 1 2 (Ω) = H 1 (Ω), note that the Bernstein inequality in particular implies that for r = 0, the space W j defined in (2.2.5) is contained in H 1 0(Ω). herefore, in the following 11

17 we may confine the discussion to the case r = 0. Furthermore, as elements we will consider exclusively d-simplices, i.e., triangles for d = 2 and tetrahedra for d = 3. Finally, as spaces P (K) we consider P k (K) for some k N >0. hat is, for some triangulations j of Ω R d into d-simplices, we consider W j := H 1 0(Ω) C(Ω) K j P k (K). (2.2.8) In order to perform efficient computations with this space we need a basis consisting of functions that have supports that are as small as possible. For this goal, we will first select a set {a i : i J j } of points in Ω such that each u W j is uniquely determined by its values at these points, and, conversely, that for each c = (c i ) i Jj R J j there is a w j W j with w j (a i ) = c i, i J j. We will call such a set a nodal set. For each K j we select the set of points S K = {x K : λ K (x) k 1 N d+1 }, (2.2.9) where λ K (x) R d+1 are the barycentric coordinates of x with respect to K, see Figures 2.2, 2.3, 2.4. Now, assuming that j is conforming, a nodal set is the union of these sets, excluding those on the boundary of Ω. Indeed, note that each p P k (K) is uniquely determined by its values in the points from the set S K. hen the statement follows from the fact that for u W j the degrees of freedom on each face of K j uniquely identify the restriction of u to that face. Given a nodal set {a i : i J j }, we define a corresponding basis {φ i : i J j } W j by { 1 if i = m φ i (a m ) = (2.2.10) 0 if i m. he set {φ i : i J j } is called a nodal basis and each w j W j can be uniquely represented as w j = i J j w j (a i )φ i. (2.2.11) In case of a possibly nonconforming triangulation, we define the nodal set as {x K j S K x / Ω and if x K j then x S K }. (2.2.12) he basis for W j is then again defined by (2.2.10). For each w C(Ω) we define the nodal interpolant I j w = i J j w(a i )φ i. (2.2.13) he following theorem gives estimates for the interpolation error w I j w ([31, 13]) 12

18 Figure 2.2: Sets S K in 2D and 3D for k = 1 Figure 2.3: Sets S K in 2D and 3D for k = 2 heorem Let ( j ) j N be a shape-regular sequence of triangulations. hen w I j w W s p (Ω) h t+1 s j w W t+1 p (Ω) for all w W t+1 p (Ω), (2.2.14) for parameters s, t and p as in (2.2.6) (and r = 0), with the additional condition t + 1 > d p. Remark Note that this theorem implies (2.2.6) for t + 1 > d. o obtain (2.2.6) p without this additional condition, one can apply so-called quasi-interpolants ([31, 14]). Finally we note that in case the domain Ω is not a polytope, in order to provide a good approximation of the boundary of Ω, isoparametric elements are used, which have curved boundaries. For more details about the construction of finite element spaces we refer the reader to the books ([13, 9, 2, 32, 10]). 2.3 Local Mesh Refinement, Adaptivity and Optimal Complexity Algorithms Approximation of a given function. Linear versus nonlinear and adaptive approximations In this section, as an example to illustrate ideas, we consider the problem of approximating a given function f : Ω R by piecewise constant functions, where Ω := [0, 1]. For 13

19 Figure 2.4: Sets S K in 2D and 3D for k = 3 W L (Ω), by e(f, W ) we denote the error in uniform norm (L -norm) of the best approximation of f by elements of W, i.e., e(f, W ) := inf w W f w L (Ω). (2.3.1) We will describe here three types of approximation, namely linear, nonlinear and adaptive. Let be a triangulation of Ω, i.e., a finite collection of essentially disjoint closed subintervals K such that Ω = K K. Let W 0 ( ) be the space of piecewise constant functions relative to, i.e., W 0 ( ) := P 0 (K). (2.3.2) K Let us first consider a uniform triangulation, meaning that all subintervals have equal length. From the estimate (2.2.6) we infer that inf f w L w W 0 (Ω) 1 u W 1 (Ω) for all f W (Ω), 1 (2.3.3) ( ) i.e., the optimal order 1 for piecewise constant approximation is attained for f W 1 (Ω). What is more, in ([18]) it is shown that e(f, W 0 ( )) α f Lip(α, L (Ω)). (2.3.4) Here, the space Lip(α, L p (Ω)), 0 < α 1, 0 < p is the space of all functions f L p (Ω) for which f( + h) f( ) Lp[0,1 h) Mh α, 0 < h < 1 (2.3.5) with the smallest M 0 for which (2.3.5) holds being the norm f Lip(α,Lp(Ω)). he space Lip(α, L (Ω)) is equal to W (Ω). 1 hanks to (2.3.4), we know precisely which class of functions can be approximated with order α by piecewise constants on uniform triangulations. his type of approximation where the triangulations are chosen in advance and are independent of the target function f is called linear. 14

20 Another type of approximation is nonlinear approximation. Nonlinear approximation allows the triangulations of Ω to depend on f. Let W 0 NL(n) :=, =n W 0 ( ), (2.3.6) i.e., the space of piecewise constants with respect to some partition of [0, 1] into n subintervals, which, clearly, is not a linear space. Whereas for linear approximation the optimal order 1 is attained for functions f Lip(1, L (Ω)), as shown in ([18]) the nonlinear approximation provides the optimal order for f Lip(1, L 1 (Ω)), i.e., e(f, W 0 NL(n)) n 1 f Lip(1, L 1 (Ω)). (2.3.7) Since Lip(1, L 1 (Ω)) is a much larger space than Lip(1, L (Ω)), with nonlinear approximation it is possible to realize the optimal approximation order for a much larger class of functions. Note, however, that the sequence of underlying optimal triangulations is not explicitly given. he third type of approximation we discuss is adaptive approximation. his is a kind of nonlinear approximation given by an algorithm that, as we will see, produces a sequence of piecewise constant approximations that converges with the optimal order under only slightly stronger conditions on f than needed for (best) nonlinear approximations. Since this is the first adaptive algorithm that we mention here, we will discuss it in some detail. Given ε, which is the target accuracy of the approximation, the adaptive algorithm ADAP- IVE subdivides those elements K for which the error inf P0 (K) R f c L (K) ε into two equal parts until the desired tolerance is met. Algorithm 2.1 Adaptive Algorithm ADAPIVE[f, ε, K] /* Called with K = [0, 1], the output of the algorithm is the triangulation such that e(f, W 0 ( )) ε. */ if inf c R f c L (K) > ε then Subdivide K into two equal parts K 1 and K 2. 1 :=ADAPIVE[f, ε, K 1 ] 2 :=ADAPIVE[f, ε, K 2 ] := 1 2 endif In [18] it was shown that this adaptive algorithm attains the optimal approximation order 1 if f satisfies the condition f L log L, where L log L stands for the space of all integrable functions g for which g L log L := g(x) (1 + log g(x) )dx <. (2.3.8) Ω his space is only slightly smaller than Lip(1, L 1 (Ω)), membership of which was needed for convergence of order 1 with best partitions. As an illustration, we consider the following example. 15

21 1 Nonlinear approximation for y=x 1/ y x Figure 2.5: Example Let us consider the function f(x) = x α with 0 < α < 1. Since it belongs to Lip(α, L (Ω)), whereas it doesn t belong to a higher order Lipschitz space, in view of (2.3.4), this function can be approximated on uniform triangulations with order exactly α, i.e., e(f, W 0 ( )) α. (2.3.9) In particular, the optimal order 1 can not be attained by linear approximation. Noting that f Lip(1, L 1 (Ω)), we conclude that this function can be approximated with order 1 using nonlinear approximation. Indeed, choose the points x k := (k/n) 1/α, k = 0, 1,..., n, which are preimages of y k := (k/n), k = 0, 1,..., n (see Fig. 2.5, Fig. 2.6). hen, with = {[x i, x i+1 ]} n 1 i=0 one easily verifies that e(f, W 0 NL(n)) e(f, W 0 ( )) 1/n, (2.3.10) i.e., with nonlinear approximation the optimal order 1 is attained for any 0 < α < 1. Finally what is more, since for any α (0, 1), f L log L, this optimal order is also attained by the algorithm ADAPIVE Singular solutions of boundary value problems In the previous section, we discussed the problem of approximating a given function. he situation is more complicated if we want to construct an algorithm for the numerical solution of a boundary value problem, since the solution is unknown and can have a singularity somewhere in the domain Ω. he origin of a singularity can be a concentration 16

22 1 Nonlinear approximation for y=x 1/ y x Figure 2.6: of stresses in elasticity, boundary/internal layers in environmental pollution problems, or a propagation of shock-waves in fluid mechanics ([4, 20, 33]). As an example, let us consider the Poisson problem (2.2.1) in two space dimensions, where Ω is a polygon (Fig. (2.7)), and the right-hand side is sufficiently smooth. hen it is known ([2]) that the regularity of the solution u depends on the maximum interior angle ω at the corners in the boundary of the domain. More precisely, for any ε > 0 we have u H π/ω+1 ε (Ω), whereas generally u / H π/ω+1 (Ω). (2.3.11) Consider now the finite element spaces W j from (2.2.5) of order k + 1 with respect to a sequence of quasi-uniform triangulations ( j ) j N. From (2.2.6), with N j := j h 2 j we infer that inf u w j H w j W 1 (Ω) N k/2 j u H k+1 (Ω) for all u H k+1 (Ω) (2.3.12) j We conclude that the optimal order k/2 is attained if u H k+1 (Ω). Actually, one can show that this is sharp, so that from (2.3.11) we see that the rate of such finite element approximations is not restricted by the regularity of the solution only if ω π/k. For k 4, there is no polygon that satisfies this condition. For k = 3, it is only satisfied for an equilateral triangle, and for k = 1, it requires that there are no re-entrant corners. o handle the case that ω > π/k, algorithms based on adaptive triangulations are often proposed with the aim to retain the optimal rate of convergence. We are interested in the question whether such adaptive procedures can really lead to a significant improvement of the numerical efficiency. An example of a typical adaptive finite element mesh for the problem with a corner singularity is given in Fig

23 Figure 2.7: Polygon with maximum interior angle ω Figure 2.8: ypical adaptive finite element mesh for a problem with a corner singularity 18

24 2.3.3 Nonconforming triangulations and adaptive hierarchical treestructures As we have described before, finite element spaces consist of suitable subspaces of all piecewise polynomials of a certain degree with respect to a subdivision of Ω into a large number of tiny polytopes. Here, we will consider such approximation spaces based on subdivisions into triangles, so in two space dimensions. In this section, we describe the type of triangulations that we shall use throughout this chapter. Let be a triangulation of Ω. Recall from Definition the concepts of conforming and nonconforming triangulations and hanging and non-hanging vertices of. We shall say that an edge l of a triangle K is an edge of the triangulation if it doesn t contain a hanging vertex in its interior. he set of all edges of will be denoted by Ē, and the set of all non-hanging vertices of by V. he set of all interior edges of and the set of all interior non-hanging vertices of will be denoted by E and V respectively. If two triangles K 1, K 2 share some edge l E, they will be called edge neighbors, and if they share a vertex we will call them vertex neighbors. An adaptive algorithm will produce generally locally refined meshes. In view of the importance of having shape-regular triangulations, the question arises which local refinement strategies preserve shape regularity. he following three methods are known to do so: Regular or red-refinement. he red refinement strategy consists of subdividing a triangle by connecting the midpoints of its edges. Clearly, since we start with a nondegenerate triangulation 0, subdivision based on red refinement produces a shape regular family of triangulations. Longest edge bisection. his strategy subdivides a triangle by connecting the midpoint of the longest edge with the vertex opposite to this edge. In ([34]) it was proven that in each triangulation from any sequence 1..., n built by the longest edge bisection, the smallest angle is larger than or equal to half of the smallest angle in the initial triangulation 0. Newest vertex bisection. his strategy starts with assigning to exactly one vertex of each triangle in the initial triangulation 0 the newest vertex label. hen if at some step of the adaptive algorithm triangle K should be subdivided, we perform this by connecting the newest vertex of K with the opposite edge, and mark the midpoint of this edge as the newest vertex for both new triangles produced by this subdivision. In ([28]) it was proven that the refinement based on newest vertex bisection builds a family of triangulations which is shape regular. In this chapter we restrict ourselves to triangulations that are created by (recursive), generally non-uniform red-refinements starting from some fixed initial conforming triangulation 0. Generally these triangulations are generally nonconforming. For reasons that will become clear later (in Section 2.8.1), we will have to limit the amount of nonconformity by restricting ourselves to admissible triangulations, a concept that is defined as follows: 19

25 Figure 2.9: riangle and its red-refinement. Figure 2.10: An example of admissible triangulation. Definition A triangulation is called an admissible triangulation if for every edge of a K that contains a hanging node in its interior, the endpoints of this edge are nonhanging vertices. Figure 2.10 depicts an example of an admissible triangulation. Apparently local refinement produces triangulations which are generally not admissible. In the following we discuss how to repair this. o any triangle K from the initial triangulation 0 we assign the generation gen(k) = 0. hen the generation gen(k) for K is defined as the number of subdivision steps needed to produce K starting from the initial triangulation 0. A triangulation is called uniform when all triangles have the same generation. A vertex of the triangulation will be called a regular vertex if all triangles that contain this vertex have the same generation (see Fig. 2.11). Obviously, a regular vertex is non-hanging. Corresponding to the triangulation, we build also a tree ( ) that contains as nodes all the triangles which were created to construct from 0. he roots of this tree are the triangles of the initial triangulation 0. When a triangle K is subdivided, four new triangles appear which are called children of K, and K is called their parent. Similarly, we 20

26 Figure 2.11: A regular (left picture) and a non-regular although non-hanging vertex (right picture). introduce also grandparent/grandchildren relations. In case a triangle K ( ) has no children in the tree ( ), it is called a leaf of this tree. he set of all leaves of the given tree ( ) will be denoted by L( ( )). Apparently, the set of leaves L( ( )) forms the final triangulation. We will call a subtree of, denoted as, if it contains all roots of and if for any K all its siblings and ancestors are also in. Proposition [36] For any triangulation created by red-refinements starting from 0, there exists a unique sequence of triangulations 0, 1..., n with max K i gen(k) = i, n = and where i+1 is created from i by refining some K i with gen(k) = i. Let 1 = and V 1 =. he following properties are valid: n i=0 i 4 3, V i 1 V i and so V = n i=0v i \ V i 1 with empty mutual intersections, a v V i \ V i 1 is not a vertex of i 1, and so it is a regular vertex of i As we noted before, in case we refine some admissible triangulation only locally, the admissibility of the triangulation might be destroyed. If a given triangulation is not admissible (see Fig. 2.12), it can be completed to an admissible triangulation using the following algorithm. his algorithm makes use of the sequence 0,..., n corresponding to the input triangulation as defined in Proposition Algorithm 2.2 Make Admissible MakeAdmissible[ ] 1. a 0 := 0 2. Let i := 0 3. Define a i+1 as the union of i+1 and, when i n 2, the collection of children of those K a i that have a vertex neighbor in i with grandchildren in i+2 4. If i n 2 then i + + and go to step (3). Otherwise set := a n and stop. 21

27 Figure 2.12: An example of a non-admissible triangulation (left) and its smallest admissible refinement (right). Proposition [36] he triangulation a admissible refinement of. Moreover, constructed by MakeAdmissible is an a (2.3.13) Adaptive Algorithms of Optimal Complexity For ( i ) i being the family of all triangulations that can be constructed from a fixed initial triangulation 0 of Ω by red-refinements, let (W i ) i, (Y i ) i be the families of finite element approximation spaces defined by W i := {v C(Ω) K i P 1 (K) : v Ω = 0}, (2.3.14) Y i := K i P 0 (K). (2.3.15) he spaces Y i will be used for approximating the right-hand side of our equation. he reason for their introduction will become clear clear in the next section where we discuss a posteriori error estimators. Defining for any S H 1 0(Ω) and w H 1 0(Ω) e(w, S) H 1 0 (Ω) := inf w S w w H 1 0 (Ω), (2.3.16) the error of the best approximation from the best space W i with underlying triangulation consisting of n + 0 triangles is given by σ H1 0 (Ω) n (w) = inf i ( j ) j, i 0 n e(w, W i ) H 1 0 (Ω) (2.3.17) We now define classes for which the errors of the best approximations from the best spaces decay with certain rates. Definition For any s > 0, let A s (H 1 0(Ω)) = A s (H 1 0(Ω), (W i ) i ) be the class of functions w H 1 0(Ω) such that for some M > 0 σ H1 0 (Ω) n (w) Mn s n = 1, 2,.... (2.3.18) 22

28 We equip A s (H 1 0(Ω)) with a semi-norm defined as the smallest M for which (4.4.6) holds: and with norm w A s (H0 1(Ω)) := sup n s σ H1 0 n (Ω) (w), (2.3.19) n 1 w A s (H 1 0 (Ω)) := w A s (H 1 0 (Ω)) + w H 1 0 (Ω). (2.3.20) For approximating the right-hand side of our boundary value problem, in a similar way we define also the class A s (H 1 (Ω)) = A s (H 1 (Ω), (Y i ) i ) using the approximation space (Y i ) i. he goal of adaptive approximation is to realize the rates of the best approximation from the best spaces. We note here that a rate s > 1/2 only occurs when the approximated function is exceptionally close to a finite element function, and such a rate can not be enforced by imposing whatever smoothness condition. From now on we consider the case s 1/2. Below we sketch why we expect, similar to section 2.3.1, better results with adaptive approximation than with non-adaptive approximation based on uniform refinements. In order to do so, we temporarily consider the family of triangulations generated by newest vertex bisection. We expect however similar results to be valid with red refinements. Let ( i ) i be the family of all triangulations created by newest vertex bisections from an initial triangulation 0 and let X i := C(Ω) K i P 1 (K). (2.3.21) For simplicity, to illustrate the ideas, we dropped here the Dirichlet boundary conditions. Recently, in [6] it was shown that the approximation classes A s (H 1 (Ω), (X i ) i ) are (nearly) characterised by membership of certain Besov spaces: heorem (i) If u Bτ 2s+1 (L τ (Ω)) with 0 s 1/2 and 1/τ < s + 1/2 then u A s (H 1 (Ω), (X i ) i ) and u A s (H 1 (Ω),(X i ) i ) u B 2s+1 τ (L τ (Ω)) (2.3.22) (ii) if u A s (H 1 (Ω), (X i ) i ) then u B 2s+1 τ (L τ (Ω)), where 1/τ = s + 1/2. On the other hand, as is well-known, in two space dimensions and with piecewise linear approximation a rate s 1/2 is attained by approximations on uniform triangulations if and only if the approximated function belongs to H 2s+1 (Ω). he above result shows that a rate s 1/2 is achieved by the best approximation from the best spaces for any function from the much larger spaces Bτ 2s+1 (L τ (Ω)) for any τ > (s + 1/2) 1. Although these results were stated for the triangulations created by newest vertex bisections, as we said we expect the same results to be valid for the family of triangulations constructed by red-refinements. he difference between Sobolev spaces and Besov spaces is well illustrated by the DeVore diagram (Fig. 2.13). In this diagram, the point (1/τ, r) represents the Besov space Bτ(L r τ (Ω)). Since B2(L r 2 (Ω)) = H r (Ω), the Sobolev spaces H r (Ω) corresponds to the points (1/2, r). From this diagram we can observe that the larger s is, the larger is the space Bτ 2s+1 (L τ (Ω)) with τ = (s + 1/2) 1 compared to H 2s+1 (Ω). 23

29 o solve the Poisson problem (2.2.1) numerically, we need to be able to approximate the right-hand side within any given tolerance. We assume the availability of the following routine RHS. Algorithm 2.3 RHS RHS[, f, ε] [, f ] /* Input of the routine: triangulation f H 1 (Ω) ε > 0 Output: admissible triangulation and an f Y such that f f H 1 (Ω) ε */ As in [36], we will call the pair (f, RHS) to be s-optimal if there exists an absolute constant c f > 0 such that for any ε > 0 and triangulation, the call of the algorithm RHS performs such that both and the number of flops required by the call are + c 1/s f ε 1/s (2.3.23) Apparently, for a given s, such a pair can only exist if f A s (H 1 (Ω), (Y i ) i ). A realisation of the routine RHS depends on the right-hand side at hand. As it was shown in [36], for f L 2 (Ω), RHS can be based on uniform refinements. We will say that an adaptive method is optimal if it produces a sequence of approximations with respect to triangulations with cardinalities that are at most a constant multiple larger then the optimal ones. More precisely, we define Definition Suppose that the solution of the Poisson problem (2.2.1) is such that u A s (H0(Ω)) 1 for some s > 0 and there exists a routine RHS such that the pair (f, RHS) is s-optimal. hen we shall say that an adaptive method is optimal if for any ε > 0 it produces a triangulation j from the family ( i ) i with j ε 1/s ( u 1/s A s (H c1/s (Ω)) f ) and an approximation w j W j with u w j H 1 0 (Ω) ε taking only ε 1/s ( u 1/s A s (H0 1(Ω))+c1/s f ) flops. Note that in view of the assumption u A s, a cardinality of j equal to ε 1/s u 1/s A s (H 1 0 (Ω)) is generally the best we can expect. he questions that should be answered, being the topics of our discussions in the next sections, are How to construct an a posteriori error estimator that provides reliable error control during the adaptive approximation process? How to construct a convergent adaptive strategy? 24

30 r (1/2,r) (1/τ, r) B r τ (L τ ), 1/τ = r/2 (1/2,1) H 1 (Ω) (0,0) 1/τ Figure 2.13: DeVore diagram. On the vertical line are the spaces H r (Ω), and on the skew line are the spaces B r τ(l τ (Ω)) with r = 2s + 1 and τ = (1/2 + s) 1, i.e., r = 2/τ 25

31 How to ensure that our adaptive approximation is optimal in the above sense? How to design the computer implementation of the adaptive algorithm which use CPU, memory etc. also in an optimal way? 2.4 A Posteriori Error Analysis In this section we will apply the elegant technique developed by Verfürth ([41]) to construct and analyze an a posteriori error estimator for the case of nonconforming triangulations. Moreover, with this technique we will be able to prove the so called saturation property which, as we will show in the next section, is essential for realizing a convergent adaptive method. hinking of an adaptive algorithm for solving the boundary value problem (2.2.1), we draw the attention of the reader to the algorithm ADAPIVE. Note that this algorithm for approximating a given function is based on the full knowledge of the error. When solving a boundary value problem, however, we do not know the solution u and thus the error. Given the Galerkin approximation u, from (2.2.1) we can derive the following equation for the error e := u u find e H 1 0(Ω) such that (2.4.1) a(e, w) = f(w) a(u, w) for all w H 1 0(Ω). his problem is as expensive to solve as the initial one. However, we have the discrete approximation u and right hand side f H 1 (Ω) in our hands. his information can be used to create an a posteriori error estimator E(, f, u ). he important requirements on the estimator E are: Reliability. here exists a constant C U > 0 such that u u E C U E (2.4.2) Efficiency. he estimator is called efficient if there exists a constant C L > 0 such that C L E u u E (2.4.3) Remark A quantitative characterization of the estimator is given by the global effectivity index defined by E I effectivity = (2.4.4) u u E In view of the characterization u u E = a(u u, w) sup, (2.4.5) 0 w H0 1(Ω) w E 26

32 and assuming that f L 2 (Ω), using integration by parts, we write a(u u, w) = fw u w Ω Ω = {f + u }w [ u ]w (2.4.6) K K l E l ν l = R K w R l w, K K l E l where R K is an element residual and R l denotes an edge residual R K = R K (f) := (f + u ) K, (2.4.7) R l = R l (u ) := [ u ν l ]. (2.4.8) Here for l E, ν l is a unit vector orthogonal to l, and with l separating two elements K 1, K 2, [ w w w ](x) := lim (x + tν l ) lim (x tν l ) for all x l. (2.4.9) ν l t 0+ ν l t 0+ ν l is the jump of the normal derivative over l. Note here that, since we consider the case that u is linear, u = 0, so that indeed R K only depends on f. Having these quantities in our hands we are ready to set an a posteriori error estimator E(, f, w ) = [ K diam(k) 2 R K 2 L 2 (K) + l E diam(l) R l 2 L 2 (l)] 1/2. Before stating the theorem about our error estimator, we need to recall one more result, that will be invoked in the proof. In [36] a piecewise linear (quasi)-interpolant was constructed on admissible triangulations for H 1 0(Ω) functions. Its properties are stated in the following theorem. heorem Let be an admissible triangulation of Ω. hen there exists a linear mapping I : H0(Ω) 1 W such that w I w H s (K) diam(k) 1 s w H 1 (Ω K ) (w H 1 0(Ω), s = 0, 1, K ) (2.4.10) w I v L2 (l) diam(l) 1/2 w H 1 (Ω Kl ) (w H 1 0(Ω), l E ) (2.4.11) where for each K, Ω K is the union of uniformly bounded collection of triangles in, among them K, which is simply connected and satisfies diam(ω K ) diam(k), and where for l E, K l is a triangle that has edge l. 27