Convergence of an adaptive hp finite element strategy in higher space-dimensions

Transcription

1 Convergence of an adaptive hp finite element strategy in higher space-dimensions M. Bürg W. Dörfler Preprint Nr. 1/6 INSTITUT FÜR WISSENSCHAFTLICHES RECHNEN UND MATHEMATISCHE MODELLBILDUNG

2 Anschriften der Verfasser: Dipl.-Math. Markus Bürg Institut für Angewandte und Numerische Mathematik arlsruher Institut für Technologie IT D-7618 arlsruhe Prof. Dr. Willy Dörfler Institut für Angewandte und Numerische Mathematik arlsruher Institut für Technologie IT D-7618 arlsruhe

3 Convergence of an adaptive hp finite element strategy in higher space-dimensions M. Bürg, W. Dörfler arlsruhe Institute of Technology, Department of Mathematics, D-7618 arlsruhe Abstract We show convergence in the energy norm for an automatic hp-adaptive refinement strategy for finite elements on the elliptic boundary value problem. The result is a generalization of a marking strategy, which was proposed in one space-dimension, to problems in two- and three-dimensional spaces. eywords: adaptive mesh refinement, a posteriori error estimates, convergence, hp version of the finite element method 1. Introduction The finite element method is a widely accepted tool for the numerical solution of partial differential equations. Nowadays a posteriori error estimation is an expected and assessed feature in scientific computing. It is used for adaptively creating approximation spaces and to assess the accuracy of numerical solutions. The performance of the method can be improved by mesh refinement h-refinement or the use of higher order ansatz spaces p-refinement. Taking a combination of both hp -refinement can lead to exponentially fast convergence with respect to the number of degrees of freedom [19]. For the h-fem adaptive mesh creation is discussed in e. g. [1, ]. For the p- and hp -FEM several strategies have been proposed to create problem-dependent meshes adaptively. There are a lot of different approaches to decide whether it is favourable to increase the polynomial degree p or to decrease the mesh size h. E. g. in [9, 1] the analyticity of the solution is estimated, in [, 8, 1] local boundary value problems are solved and in [6, 18] the global interpolation error is minimized. In this paper we generalize the refinement strategy proposed in [8] to problems in two and three spacedimensions and present numerical results of the application of this refinement strategy to some representative problems. The paper is organized as follows: In Section we state some general assumptions and introduce the model problem. In Section we develop the refinement strategy for two- and three-dimensions and prove its convergence in two space-dimensions explicitly. Further we give numerical results of the application of our refinement strategy to some examples of the model problem in Section.. Preliminaries In this section we give the notations and general assumptions, which we will use throughout this paper. Further we state the model problem, which we will consider here. Corresponding author addresses: buerg@kit.edu M. Bürg, willy.doerfler@kit.edu W. Dörfler Preprint submitted to Elsevier July 16, 1

4 .1. Notations and Assumptions For d = 1,, let Ω R d be an open domain with polygonal boundary. By L Ω and H 1 Ω we denote as usual the Lebesgue space of square-integrable functions in Ω respectively the Sobolev space of functions in H 1 Ω with homogeneous Dirichlet boundary. By we denote a triangulation of Ω. Throughout this paper we assume that satisfies the following regularity property [19, ]. Definition 1 Shape regularity. Let be the image of the reference cell under some affine map F :. Set h := diam. Then is called γ-shape regular, if and only if there exists some γ > such that F L + h F 1 F h L γ. Let be arbitrary. Then we set and for i > 1 we inductively define ω,1 := {x L : L, and L have a common edge} ω,i := ω,i 1 For simplicity we will write ω := ω,1. Let p := p, p N, be the polynomial degree vector on mesh. Then we set p ω := max p L, L ω where p L denotes the polynomial degree of the ansatz space on cell L. Let f L Ω and q N be arbitrary. Then we denote the local L -projection of f onto the space of polynomials of degree q by f q. For d {1,, } let Q := [, 1] d be the reference square respectively the reference cube and T := { x R d : < x 1,..., x d,. } d x i 1 i=1 be the reference d-simplex. Obviously we have Q = T for d = 1. Let S p {u, Ω := H 1 Ω : u F P p } be a finite dimensional approximation space of piecewise polynomials with homogeneous Dirichlet boundary, where { span {x i y j : i, j p}, if P p = = Q span {x i y j : i + j p}, if = T. Finally, for N N we denote the mesh after refinement step N by N... The Model Problem Throughout this paper we consider the problem to find u : Ω R such that u = f in Ω, u = on Ω 1

5 for given f : Ω R. Multiplying the first equation of problem 1 with φ H 1 Ω and integrating by parts yields the weak formulation φ T u = φf φ H 1 Ω. Ω Ω If f L Ω, then the Lax Milgram Theorem states that there exists a unique u H 1 Ω satisfying. Analogously, the discrete problem reads: Find u N S p N, Ω such that φ N T u N = φ N f φ N S p N, Ω for given f L Ω. Ω Ω Remark 1 Nonhomogeneous boundary conditions. For problems with nonhomogeneous Dirichlet boundary conditions u = g on Ω, where g : Ω R, we may simply assume that there exists a lifting u g H 1 Ω satisfying u g = g on Ω such that this solution u is given by u := u + u g, where u is the solution of the homogeneous boundary value problem 1.. hp -Adaptive Refinement Strategy In this section we present our refinement strategy for problems in two and three space-dimensions and prove the convergence of this hp -adaptive algorithm for two space-dimensions explicitly..1. Refinement Strategy The refinement strategy, which we present here, was developed in [8] for one space-dimension. In this section we present a modified formulation, which is applicable to problems in two and three spacedimensions as well. Let TOL > be some prescribed tolerance for our computed solution. We start our algorithm with some coarse decomposition of Ω and the approximation space S p, Ω H 1 Ω. We solve the discrete problem on this initial discretization and check if the energy error is below the prescribed tolerance Energy error indicator We will estimate the energy error a posteriori, that is, only knowing the discrete solution u N and data. For two dimensions the following method was proposed in [16]: Definition Residual error estimator. Let N be arbitrary, h its diameter and p the polynomial degree on the cell. Then the residual error indicator η associated with element is given by η = η R, + η B,, where is the weighted residual on cell and ηr, := h p f p 1 + u N L η B, := f Ω [ ] h f un p f n f is the weighted boundary residual with n f denoting the outward pointing unit normal and [w] denoting the jump of a function w over face f. L f

6 .. Refinement Patterns A local procedure to enhance the finite element space is called a refinement pattern. For the h-adaptive FEM the common refinement pattern to choose is the equal-sized bisection in every coordinate-direction. To reduce the error contribution of the boundary terms as well, we have to assure that no new hangingnodes are produced at the edges of the cell. Thus we also have to refine the neighboring cells at least anisotropically. This is shown for a quadrilateral in Figure 1 on the left-hand side. For the p-adaptive FEM the common choice is to increase the polynomial degree p by one, see Figure 1 on the right-hand side. Also in this case we have to make sure that no new constrained degrees of freedom are created. Hence we increase the polynomial degree on the neighboring cells as well. Further we provide the choice of increasing the polynomial degree by two in some applications. This is shown in Figure. Thus for the hp -adaptive FEM we have at least two different refinement patterns, the bisection and the increment of the polynomial degree. Of course there are much more refinement patterns, which can be applied here, e. g. anisotropic h-refinement or the increment of the polynomial order by some arbitrary number. Henceforth let us assume that we have n N, n, different refinement patterns to choose from. Figure 1: Refinement patterns on cell. Left: Bisection in x- and y-direction. Right: Increase polynomial degree by one...1. Convergence indicators for refinement cases Let j {1,..., n} and N be arbitrary. Then we denote the finite element space of functions compactly supported in ω with refinement pattern j applied to by S p,,j N, ω. Let β,j be the solution of the optimization problem β,j η = sup φ S p,,j N,ω If v N,j S p,,j N, ω is the solution of the problem ω φf pω φ T u N φ L ω φ T v N,j = ω φf pω φ T u N ω φ S p,,j N, ω,.

7 Figure : Increase polynomial degree by two. then we can see that for all φ S p,,j N, ω ω φf pω φ T u N φ L ω Thus v N,j is the solution of the optimization problem. ω v N,j f pω v N,j T u N v N,j L ω = v N,j L ω.... Marking cells for refinement Hence, if we solve problem for every refinement case, we have a guess which refinement case provides the biggest error reduction on every cell. Unfortunately this is not enough, because simply heading for the biggest error reduction might be inefficient. Thus we also would like to take into account the amount of work required for the achieved reduction of the error. For this we define numbers w,j to be the number of degrees of freedom, which the local refinement space S p,,j N, ω has after we applied refinement pattern j to cell. Then we can mark the cells for refinement by looking for a solution A N, j AN of the minimization problem w,j = min 5 β,j under the constraint A N A N β,j ε θ ε. 6 Unfortunately this problem is NP-hard. Here we suggest the following strategy: First we define j N for every N by w,j w,j = min β,j j=1,...,n and then we construct a minimal possible A N fulfilling constraint 6 as proposed in [7]. β,j... hp -adaptive refinement algorithm In summary, we can state our hp -adaptive refinement algorithm as follows: 5

8 S Start with coarse grid, N :=. S1 Solve the linear system of equations to get an approximate solution u N S p N, Ω. Compute a local error indicator ε on every cell N. If ε := ε > TOL : For all cells N and all refinement cases j = 1,..., n compute the values β,j as shown in Section..1. Else: STOP S To mark the cells for refinement we look for a tuple A N, j AN as proposed in Section... S Refine the cells contained in A N according to the refinement patterns j AN and set N := N +1. Continue with step S1... A Posteriori Error Estimate An a posteriori error estimate is an upper bound of some error quantity, which only relies on information obtained from the solution of the discrete problem. Here we consider the energy error u u N L Ω and prove u u N L Ω ε, where the ε are some computable quantities that are local in. Theorem 1 A posteriori error estimates. Let ε > and N be a γ-shape regular mesh. Assume that for all 1, N with 1 holds p 1 γ p γp 1. 7 Then there exists some constant C 1 >, independent of cell diameter h and polynomial degree p for all N, such that u u N L Ω C 1 η + h p f f p 1 L with η as in Definition. Further there exists some C ɛ >, independent of h and p, such that η C ɛ p 1+ɛ u u N L ω + h f p 1 f L ω. Proof. See [16], Lemmas.1,. and.5 with α =... Convergence of hp -Adaptive Strategy p 1 ɛ Now we will show the convergence of our refinement strategy in two space-dimensions. For this we have to show that the energy error decays in every refinement step, i. e. for N N and some κ, 1 we have u u N+1 L Ω κ u u N L Ω. Before we give the proof we first introduce an hp -efficient Scott Zhang type interpolation operator, which was developed in [16] and is used in our proof later on. This interpolation operator is a generalization of the interpolation operator of [] to the hp -context. The following theorem states an upper bound for the interpolation error. 6

9 Theorem Scott Zhang type interpolation. Let be a γ-shape regular triangulation of Ω. Assume that assumption 7 holds for all 1, with 1. Then there exist a linear operator I : H 1 Ω S p, Ω and a constant C I >, depending only on γ, such that 1. Iu Ω = u Ω =. For all holds u Iu L ω C I h p u L ω,5. Proof. In the vertex-based formulation and notation of Theorem. in [16] we have the two inequalities u Iu L ω V V u Iu L ω V,1 and h V p V u L ω V, h V p V u L ω,5 C h p u L ω,5, where C depends solely on γ. Thus the result is a direct consequence of the cited theorem. Now we are ready to prove the convergence of our algorithm. This theorem is a generalization of Theorem of [8] to two space-dimensions. The proof for the three-dimensional case follows the same arguments, but with three-dimensional versions of Theorems 1 and. For an idea on their proofs see [15]. Theorem Convergence. Let N N be arbitrary and S p N, Ω be the finite element space over a γ-shape regular discretization N of a bounded domain Ω R. Further we assume that inequality 7 holds for all 1, N with 1. Let u H 1 Ω be the solution of problem and u N S p N, Ω be the solution of the discrete problem in iteration step N. Let θ [, min{1, C 1 }] and ε > 1/ such that θ 8C 1C cov C max p ε 8 with C 1 from Theorem 1, C cov from 1 and C > depending only on γ. Further we assume that the data error is controlled by the discretization error, i.e. there exists some µ, min 1, 1 C εc cov max p ε+ 1 independent of h, p, N, where C ε is from Theorem, such that h p f f p 1 L µ η. 9 In addition we assume that in every refinement step N N the minimization problem 5, 6 has a solution A N, j AN. Let u N+1 S p N+1, Ω be the solution of the discrete problem in the following iteration step N + 1. Then there exists a constant κ, 1 such that u u N+1 L Ω κ u u N L Ω. Thus, if we run the hp-adaptive algorithm presented in Section.., the energy error decreases uniformly in every iteration step. 7

10 Proof. Since S p N, Ω S p N+1, Ω, we can use the Galerkin orthogonality u u N+1 T u N+1 u N = to get Ω u u N L Ω = u u N+1 L Ω + u N+1 u N L Ω. Thus the proof follows, if there exists some κ, 1 such that u N+1 u N L Ω 1 κ u u N L Ω. 1 Let N and v N+1 S p N+1, Ω with supp v N+1 ω. Then by Galerkin orthogonality we see v N+1 T u N+1 u N = v N+1 T u N+1 u + u u N ω ω = v N+1 f v N+1 T u N ω = v N+1 f pω v N+1 T u N + v N+1 f f pω. ω ω Using the definition of f pω we get v N+1 T u N+1 u N ω = v N+1 f pω v N+1 T u N + v N+1 v N f f pω, ω ω which implies v N+1 T u N+1 u N ω v N+1 f pω v N+1 T u N v N+1 v N f f pω ω ω, and applying the Cauchy Schwarz inequality yields v N+1 f pω v N+1 T u N ω L v N+1 L ω u N+1 u N L ω + v N+1 v N L ω f f pω. 11 ω Choosing v N := I N v N+1 with I N : H 1 Ω S p N, Ω as in Theorem implies v N+1 v N L ω = v N+1 I N v N+1 L ω C I h p v N+1 L ω,5. Since supp v N+1 ω ω,5 we have v N+1 v N L ω C I h p v N+1 L ω 8

11 and inserting into 11 yields v N+1 f pω v N+1 T u N ω u N+1 u N L ω + C I h p f fpω L ω Dividing by v N+1 L ω and taking the supremum on both sides yields ω sup φf pω φ T u N φ S p,,j N,ω φ L ω ω v N+1 f pω v N+1 T u N v N+1 L ω and with we have sup v N+1 S p N+1,Ω supp v N+1 ω u N+1 u N L ω + C I h p f fpω L ω h f L β,j η u N+1 u N L ω + C I fpω. p ω Squaring both sides and summing up over N yields β,j η β,j η A N u N+1 u N L ω + C I C cov u N+1 u N L Ω + C h p v N+1 L ω. f f pω L ω h p f f p 1 L use the minimal property of f pω in L ω with the covering constant C cov := max {L N : L ω }, 1 where A denotes the cardinality of the set A, and C > is a constant that depends only on γ. Finally by assumption 9 we get β,j η C cov u N+1 u N L Ω + C µ η. 1 A N From Theorem 1 we know that there exists some C 1 > such that u u N L Ω C 1 η + h p f f p 1 L C µ C 1 η 9 η

12 by 9 and µ 1. Multiplying both sides by θ [, 1] yields θ u u N L Ω C 1θ η and by applying inequality 6 with ε := η we have θ u u N L Ω C 1 β,j η. A N Then we see θ u u N L Ω C 1C cov u N+1 u N L Ω + C µ η 1 by equation 1. From Theorem 1 we know η C ε p ε+1 u u N L ω + h p 1 ε f f p 1 L ω C εc cov max p ε+1 u u N L Ω + max p ε+1 h N p f f p 1 L and assumption 9 yields Hence for we have η C εc cov max p ε+1 u u N L Ω + µ max p ε+1 1 µ < C εc cov max p ε+ 1 C εc cov max p ε+1 η N 1 µ C εc cov max p ε+1 u u N L Ω N and with the even more restrictive assumption we get Inserting into 1 yields θ u u N L Ω 1 µ < C εc cov max η C εc cov max p ε+1 p ε+ 1 η 15 u u N L Ω. C 1 C cov u N+1 u N L Ω + µ C εc cov C max p ε+1 u u N L Ω. 1

13 and by 15 we get Therefore θ u u N L Ω C 1C cov u N+1 u N L Ω + C max and by 8 we have p ε 1 u u N L Ω. 1 θ C 1C cov C C 1 C cov max p ε 1 u u N L Ω u N+1 u N L Ω θ C 1C cov C max p ε 1 1 θ. Since C cov > 1 and θ < C 1, inequality 1 holds with This completes the proof of the theorem. κ := 1 θ C 1 C cov. Remark. 1. For θ [, min{1, C 1 }] assumption 8 can always be satisfied by choosing p big enough for some N. However, the qualitative convergence result might be a first step towards proving optimality [5, 1].. The data saturation assumption 9 can only be satisfied, if the integrals on the left-hand side are computed with negligible error. To achieve this one can use high-order quadrature rules and perform local refinement according to the local error indicator h p f f p 1 L until 9 is satisfied or build some data error control into the whole algorithm as proposed in [1, 17].. If the computed values β,j are too small or θ is chosen too large, then the solution A N, j AN of the minimization problem 5, 6 might not exist for some N N. Especially this is the case if max A N β,j < θ. Then our algorithm will continue with global refinement, but a uniform decrease of the error cannot be guaranteed anymore.. If β,j is uniformly bounded from below, then there is a choice for θ such that convergence according to Theorem is assured. We think this can be shown following the ideas in [8], Theorem 5. However, this property can also be assured by monitoring the known values of β,j thus checking the convergence in an a posteriori way.. Numerical Results We apply our refinement strategy to some representative problems of the form 1. For both, two and three space-dimensions, we choose one example with a smooth solution and one with a singular solution to show how the refinement strategy behaves under these very different circumstances. The computations are performed with the finite element library deal.ii [, ] in combination with parts of the Trilinos Project [1, 11]. 11

14 .1. Example 1 In the first example our algorithm is applied to a problem of the form 1 with a smooth solution. Let Ω :=, 1, g := and ux, y := x1 xy1 y1 y exp.5x 1. We start our algorithm with the triangulation consisting of cells of same size. As refinement patterns we offer the usual bisection in every coordinate-direction of a cell as h-refinement and the increment of the polynomial degree of the cell by one as p-refinement. We denote this strategy by Θ 1. As initial polynomial degree we choose p = on all cells. For θ [, 1] we choose θ :=.5. In Figure we plot the number of degrees of freedom vs. the true energy error respectively the estimated error in log 1 -log 1 -scale on the left-hand side and the distribution of the polynomial degree on the final grid on the right-hand side. In Table 1 on the left-hand side the marking history is shown. There we denote the number of cells marked for h-refinement by #h and the number of cells marked for p-refinement by #p. We observe that no h-refinement takes place and the algorithm performs full p-refinement in every refinement step, which is the case because the solution u is very smooth. On the final grid the polynomial degree is equally distributed with value 8. In Figure we plot θ vs. the number of degrees of freedom on the left-hand side and θ vs. the number of refinement steps on the right-hand side. There we can see that the minimal number of degrees of freedom we achieved is 1967, which is attained for θ =.1. For θ.5 global refinement is performed. The number of refinement steps decays for increasing θ. In Figure 5 on the left-hand side we plot θ vs. the computation time in seconds. The minimal computation time we achieved is approx s. This supports our choice θ =.5. For we have β,j [.7,.58]. Thus the optimal value for θ lies near the minimal value of β,j. Figure : Example 1: Strategy Θ 1. Left: Number of degrees of freedom vs. error. Right: Distribution of polynomial degree. In a second try we add the increment of the polynomial degree by two as a third refinement possibility to the algorithm. We denote this refinement strategy by Θ. In Figure on the left-hand side we plot the number of degrees of freedom vs. the true energy error respectively the estimated error in log 1 -log 1 - scale. We note that this strategy produces the same final result as Θ 1, but with half the number of refinement steps. In Figure we plot θ vs. the number of degrees of freedom and θ vs. the number of refinement steps. The minimal number of degrees of freedom we achieved for this run is 1 for θ =.5. The number of refinement steps is reduced significantly with this third refinement pattern in use and the computation time is reduced to approx. 1.6 s. 1

15 Step #Cells maxp #h #p Step 1 5 #Cells maxp #h 9 #p 8 Table 1: Left: Example 1. Marking history. Right: Example. Marking history. Figure : Example 1. Left: θ vs. number of degrees of freedom. Right: θ vs. number of refinement steps... Example The second example shows how the algorithm performs, if the solution of model problem 1 has a corner singularity. Let Ω := 1, 1 \ [, 1 1, ], f := and ur, φ := r sin φ, where r [, 1 and φ [, π denote the polar coordinates, and g := u. Again we use refinement strategy Θ1. We start our computations with initial grid consisting of 8 cells of the same size and polynomial degree p = on every cell. We choose θ =.5. In Figure 5, on the right-hand side, we plot the number of degrees of freedom vs. the true energy error respectively the estimated error in log1 -log1 -scale. The distribution of the polynomial degree on the final grid is shown in Figure 6 on the left-hand side. We see that the algorithm performs h-refinement around the singularity located at only. The polynomial degree is raised up to 8 away from the singularity. In Table the marking history is shown. We observe that the algorithm starts with some h-refinements around the singularity and then performs more and more p-refinements. In Figure 6 we plot θ vs. the number of degrees of freedom on the right-hand side. There we can see that the number of degrees of freedom increases for increasing θ until θ =.. For θ. the number of degrees of freedom stays constant at its maximum value 157, since in this range global refinement is performed. The minimal number of degrees of freedom we achieved is 69 for θ =.15. θ vs. the number of refinement steps is shown in Figure 7 on the left-hand side. We observe that the number of refinement steps decays for increasing θ. Further we see that the number of 1

16 Figure 5: Left: Example 1. θ vs. computation time. Right: Strategy Θ. Number of degrees of freedom vs. error. degrees of freedoms and the number of refinement steps change discontinuously in the interval [.5,.]. This behaviour can also be observed in the computation time, which is required for the different values of θ see Figure 7 on the right-hand side. The minimal computation time is achieved for θ =.5 and we see that with β,j [.18, 1] for we have a good indicator for the choice of θ again. Figure 6: Example. Left: Distribution of polynomial degree. Right: θ vs. number of degrees of freedom... Example Now we investigate the performance of our hp -adaptive refinement strategy in three space-dimensions. First we consider a smooth example again. Let Ω := [, 1], ux, y, z := sinπx sinπy sinπz, f := π u and g :=. We use refinement strategy Θ1 and start our computations with initial grid consisting of equally sized cells. As initial polynomial degree we choose p = on all cells. We choose θ =.. In Figure 8 we plot the number of degrees of freedom vs. the true energy error respectively the estimated error in log1 -log1 -scale on the left-hand side. We can see that the polynomial degree equals 8 on all cells except the ones located at the corners. There the polynomial degree is raised to 7 only. In Table 1 on the right-hand side the marking history is shown. We observe that the algorithm performs p-refinements 1

17 Step #Cells maxp h p Step #Cells maxp h p Table : Example. Marking history Figure 7: Example. Left: θ vs. number of refinement steps. Right: θ vs. computation time. only. In Figure 9 we plot θ vs. the number of degrees of freedom on the left-hand side. There we can see that the number of degrees of freedom stays constant unless a jump at θ =.5. This is due to the point-wise symmetry of the analytic solution. For θ. the number of degrees of freedom is and for θ. the number of degrees of freedom is 89. The number of refinement steps has constant value 5 for all θ [, 1]. In Figure 7 on the right-hand side we plot θ vs. the computation time in seconds. For we have β,j [.7,.6]. Hence the optimal value for θ lies again near the minimal value of β,j. In a second try we use the refinement strategy Θ. We plot the number of degrees of freedom vs. the true error in Figure 1 on the left-hand side. The distribution of the polynomial degree is shown on the right-hand side. We observe that also strategy Θ raises the polynomial degree to 8 on all cells except the ones located at the corners. There the polynomial degree has value 6 only... Example In our last example we consider again a problem with a corner singularity. Let Ω := [ 1, 1] \, 1], ux, y, z := x + y + z 1, f x, y, z := 1 9 x + y + z 15

18 Figure 8: Example : Θ1. Left: Number of degrees of freedom vs. error. Right: Distribution of polynomial degree. Figure 9: Example. Left: θ vs. number of degrees of freedom. Right: θ vs. computing time. and g := u. Again we use refinement strategy Θ1 and start our computations with initial grid consisting of 8 equally sized cells and polynomial degree p = on all cells. We choose θ =.16. In Figure 11 on the left-hand side we plot the number of degrees of freedom vs. the true energy error respectively the estimated error in log1 -log1 -scale. In Table we show the marking history. We observe that the algorithm starts with some h-refinements around the singularity located at and then performs more and more p-refinements. In Figure 11 we plot θ vs. the number of degrees of freedom. There we can see that the number of degrees of freedom is increasing for θ.1. Then the number of degrees of freedom stays constant at The minimal number of degrees of freedoms we achieved is 519 for θ =.16. For θ.16 the number of refinement steps is constant as can be seen in Figure 1. Only for θ =.16 the algorithm requires more refinement steps. For we have β,j [.15, 1]. Also in this example the minimal value of β,j is a good indicator for the choice of θ. 16

19 Figure 1: Example : Θ. Left: Number of degrees of freedom vs. error. Right: Distribution of polynomial degree. Figure 11: Example : Left: Number of degrees of freedom vs. error. Right: θ vs. number of degrees of freedom. [1] M. Ainsworth and T.J. Oden. A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York,. [] M. Ainsworth and B. Senior. An adaptive refinement strategy for hp-finite element computations. Appl. Numer. Math., 6: , [] W. Bangerth, R. Hartmann, and G. anschat. deal.ii Differential Equations Analysis Library, Technical Reference. [] W. Bangerth, R. Hartmann, and G. anschat. deal.ii a general-purpose object-oriented finite element library. ACM Trans. Math. Softw.,, 7. [5] P. Binev, W. Dahmen, and R. DeVore. Adaptive finite element methods with convergence rates. Numer. Math., 97:19 68,. [6] L. Demkowicz, W. Rachowicz, and Ph. Devloo. A fully automatic hp-adaptivity. J. Sci. Comp., 171 :17 155,. 17

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21 [15] J.M. Melenk. hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation. SIAM J. Numer. Anal., 1:17 155, 5. [16] J.M. Melenk and B.I. Wohlmuth. On residual-based a posteriori error estimation in hp-fem. Adv. Comp. Math., 15:11 1, 1. [17] P. Morin, R.H. Nochetto, and.g. Siebert. Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal., 8:66 88,. [18] W. Rachowicz, D. Pardo, and L. Demkowicz. Fully automatic hp-adaptivity in three dimensions. Comp. Meth. Appl. Mech. Eng., 195:816 8, 6. [19] Ch. Schwab. p- and hp-finite Element Methods. Clarendon Press, Oxford, [] L.R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 5:8 9, 199. [1] R. Stevenson. Optimality of a standard adaptive finite element method. Found. Comput. Math., 7:5 69, 7. [] B. Szabó and I. Babuška. Finite Element Analysis. Wiley, New York, [] R. Verfürth. A review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner, Chichester,

22 IWRMM-Preprints seit 8 Nr. 8/1 Nr. 8/ Nr. 8/ Nr. 8/ Nr. 8/5 Nr. 8/6 Nr. 8/7 Nr. 8/8 Nr. 8/9 Nr. 8/1 Nr. 8/11 Nr. 8/1 Nr. 9/1 Nr. 9/ Nr. 9/ Nr. 9/ Nr. 1/1 Nr. 1/ Nr. 1/ Nr. 1/ Nr. 1/5 Nr. 1/6 Patrizio Neff, Antje Sydow, Christian Wieners: Numerical approximation of incremental infinitesimal gradient plasticity Götz Alefeld, Zhengyu Wang: Error Estimation for Nonlinear Complementarity Problems via Linear Systems with Interval Data Ulrich ulisch : Complete Interval Arithmetic and its Implementation on the Computer Armin Lechleiter, Andreas Rieder: Newton Regularizations for Impedance Tomography: Convergence by Local Injectivity Vu Hoang, Michael Plum, Christian Wieners: A computer-assisted proof for photonic band gaps Vincent Heuveline, Peter Wittwer: Adaptive boundary conditions for exterior stationary flows in three dimensions Jan Mayer: Parallel Algorithms for Solving Linear Systems with Sparse Triangular Matrices Ulrich ulisch : Begegnungen eines Mathematikers mit Informatik Tomas Dohnal, Michael Plum, Wolfgang Reichel: Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation Götz Alefeld: Verified Numerical Computation for Nonlinear Equations Götz Alefeld, Zhengyu Wang: Error bounds for complementarity problems with tridiagonal nonlinear functions Tomas Dohnal, Hannes Uecker: Coupled Mode Equations and Gap Solitons for the D Gross-Pitaevskii equation with a non-separable periodic potential Armin Lechleiter, Andreas Rieder: Towards A General Convergence Theory For Inexact Newton Regularizations Christian Wieners: A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing Arne Schneck: Constrained Hardy Space Approximation Arne Schneck: Constrained Hardy Space Approximation II: Numerics Ulrich ulisch, Van Snyder : The Exact Dot Product As Basic Tool For Long Interval Arithmetic Tobias Jahnke : An Adaptive Wavelet Method for The Chemical Master Equation Christof Schütte, Tobias Jahnke : Towards Effective Dynamics in Complex Systems by Markov ernel Approximation Tobias Jahnke, Tudor Udrescu : Solving chemical master equations by adaptive wavelet compression Christian Wieners, Barbara Wohlmuth : A Primal-Dual Finite Element Approximation For A Nonlocal Model in Plasticity Markus Bürg, Willy Dörfler: Convergence of an adaptive hp finite element strategy in higher space-dimensions Eine aktuelle Liste aller IWRMM-Preprints finden Sie auf:

23 ontakt arlsruher Institut für Technologie IT Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung Prof. Dr. Christian Wieners Geschäftsführender Direktor Campus Süd Engesserstr arlsruhe iwrmm-sekretariat@math.uka.de Herausgeber arlsruher Institut für Technologie IT aiserstraße arlsruhe Juli 1