CRi+1 and restriction operators Ii i+1 : CRi+1 7? ! CRi, which are investigated by. Braess/Verf urth. Once the hierarchical basis Hi with the basis functions. H.
Wohlmuth, B.
Adaptive Multilevel Methods for Nonconforming Finite Element Discretizations We consider adaptive multilevel techniques for nonconforming nite element discretizations of second order elliptic boundary value problems. In particular, we will focus on two basic ingredients of an ecient adaptive algorithm. The rst one is the iterative solution of the arising linear system by preconditioned conjugate gradient methods and the second one is an a posteriori error estimator for the global discretization error.
1. Introduction We consider the following linear self-adjoint second order boundary value problem with homogeneous Dirichlet conditions on the polygonal domain IR2 with boundary @ : Lu(x) := ?r (a(x)ru(x)) + b(x)u(x) = f (x); x 2 ; (1) u(x) = 0; x 2 ? := @
We assume f 2 L2 ( ) and a() resp. b() to be a piecewise continuous 2 2 matrix resp. a scalar function on
satisfying a T T a(x) a T ; x 2 ; 2 R2; a > 0 and 0 b(x) b; x 2 : For the discretization we choose the lowest order Crouzeix-Raviart elements, which are piecewise linear and continuous at the midpoints of the edges. The associated nite element space is denoted by CRh and the bilinear form on CRh CRh by ah (; ). Now we can provide the discrete variational formulation, which is uniquely solvable Rdue to the Lax-Milgram Lemma. P We are looking for an element uh 2 CRh satisfying ah (uh ; vh ) = (f; vh ) := 2Th f vh dx for all vh 2 CRh .
2. Multilevel Preconditioners
Multilevel preconditioners are based on the nestedness of the conforming nite element spaces. In the nonconforming setting we have the problem that the sequence of nite element spaces is non-nested. For more details we refer to [2], where also the references can be found. HB-type preconditioner: This preconditioner is due to Oswald, who constructs a hierarchical basis Hi by means of prolongation Iii+1 : CRi 7?! CRi+1 and restriction operators Iii+1 : CRi+1 7?! CRi, which are investigated by Braess/Verfurth. Once the hierarchical basis Hi with the basis functions iH is established, the implementation follows the same lines X as in the conforming case. CHB r = (2) aj ( iH ; iH )?1 (r; iH ) iH iH 2Hj
BPX-type preconditioner: The construction of the preconditioner is based on the following sequence of nested nite element spaces, S0L S1L : : : SjL CRj ; where SkL denotes the linear conforming ansatz space and is due
to Bramble/Oswald/Zhang. The preconditioner consists of two parts, where the rst one is exactly the same as in the conforming case. For the condition number it holds = O (1). CBPX r =
j X
X
k=0 'ki 6='ki ?1
aj ('ki ; 'ki)?1 (r; 'ki )'ki +
X
n i 2Nj
aj ( in ;
n ?1 n n i ) (r; i ) i ;
(3)
denotes the nodal nonconforming basis functions and 'ki stands for the nodal conforming ones of SkL . Alternative HB-type preconditioner: We consider a ctitious triangulation Tj+1 which is obtained from Tj by uniform re nement. Further, we need a pseudo-interpolation operator P : CRj 7?! SjL+1 , which preserves the energy norm: jkvjk jkP vjk jkvjk; where , are positive constants independent of j and a pseudo-inverse J : SjL+1 7?! CRj satisfying JP v = v; v 2 CRj . Altogether we receive the algebraic representation for CAHB . conform J T r; (4) CAHB r = J CHB conform T where CHB denotes the conforming HB-preconditioner for level j +1. Note that the computation of J v requires no additional operation. By means of the ctitious ? domain lemma the condition number estimates are inherited from the conforming case and we obtain = O j 2 .
where
n i
3. Element-oriented Error Estimator We denote by uNC the available nonconforming nite element approximation. The rst step is to consider a localized error equation with respect to a higher dimensional ansatz space. We de ne e1Q so that e1Q j 2 P 2( ) satis es Z
Z
@
ah (e1Q j ; v) = l (v ? Iv) with l (v) := (f ? L(uNC )) v dx + 1=2
a
@uNC v d; v 2 P 2( ): (5) @n J
As usual I stands for the local Lagrange interpolation operator and []J denotes the jump across @ . The computation of e1Q requires the solution of a 6 6 linear system on each inner element. We can reduce the computational amount by means of the elliptic projection e2Q of e1Q onto the three dimensional subspace of quadratic bubbles associated with the midpoints of the edges. In a nal reduction step we neglect the coupling between the bubbles. The local sti ness matrices and their diagonal parts are spectral equivalent with constants independent of . Altogether e3Q requires elementwise the solution of three scalar equations. However, e3Q ignores the discontinuity of uNC across the edges. So we only obtain a lower bound for the energy norm of the global discretization error. For a two-sided error estimator we have to introduce a suitable second term QuNC ? uNC , where Q represents a quasi-interpolation operator with the property Qv = v for all continuous v 2 CRj . Theorem: Under appropriate assumptions there holds (6) cjku ? uNC jk jke3Q + QuNC ? uNC jk + jkuNC ? uh jk 3 (7) jkeQ + QuNC ? uNC jk c(jku ? uNC jk + jkuNC ? uh jk);
where c and c are positive constants depending only on the shape regularity of 2 T0 and on the ellipticity of the problem but not on the re nement level.
See [1] for the assumptions and a detailed proof.
4. Numerical Results We consider the test problem raru = 0, where a is piecewise constant with alternating values 1 and 100. The Implementation is based on the nite element code KASKADEpversion 1.9. To test the quality of the preconditioners we start the conjugate gradient method with uo = 0 and use rt Hr 1:e ? 6 as stopping criterion where r denotes the residual vector and H the matrix of the preconditioner. In the multilevel algorithm we obtain the start vector on level i by extrapolation of the level i ? 1 solution. The accuracy is level dependent and takes into account the energy norm of the estimated error,Ni =Ni+1, where Ni denotes the number of nodes of Ti and a safety factor. In addition to the simple mean value re nement strategy we take the area of the triangles into account. This results in some sort of an extrapolation strategy. The following gure shows the performance of the preconditioners as well as the triangulations of level 3 and 7 based on the element-oriented error estimator. Note that the AHB-type preconditioner is the most ecient, because one cg-step is much cheaper than in the BPX-type case. Test Problem - eps = 1.e-06
Number of cg-iterations
120 HB BPX AHB
100
80
60
40
20
0 0
5000 10000 15000 20000 25000 30000 35000 40000 45000 Number of nodes
5. References 1 R. H. W. Hoppe, B. Wohlmuth: Element-oriented and Edge-oriented Local Error Estimators for Nonconforming Finite Element Methods, Mathematisches Institut, Technische Universitat Munchen, TUM-M9206 (1992) eingereicht bei: M 2 AN , Modelisation Mathematique et Analyse Numerique 2 B. Wohlmuth, R. H. W. Hoppe: Multilevel Approaches to Nonconforming Finite Element Discretizations of Linear Second Order Elliptic Boundary Value Problems, Mathematisches Institut, Technische Universitat Munchen, TUM-M9320 (1993) erscheint in: Journal of Computation and Information
Address: Wohlmuth, B., Technische Universitat Munchen, Mathematisches Institut, Arcisstrae 21, D-80290
Munchen. The author is supported by FORTWIHR: Bavarian Consortium for High-Performance Scienti c Computing.
