Hierarchical bases for elliptic problems

Abstract

Linear systems of equations with positive and symmetric matrices often occur in the numerical treatment of linear and nonlinear elliptic boundary value problems. If the CG algorithm is used to solve these equations, one is able to speed up the convergence by preconditioning. The method of preconditioning with hierarchical basis has already been considered for the Laplace equation in two space dimensions and for linear conforming elements. In the present work this method is generalized to a large class of conforming and nonconforming elements. 0. Introduction Consider elliptic problems in variational form of order 2m. Let X := HTM'2(CI), and Q. £ W be a bounded domain. We want to find u £ X such that (0.1) a(u,v):= i A[Vmu,Vmv] = F(v) Vv £ X. Ja Here, A denotes a strictly positive (possibly x-dependent) matrix; F is a functional on X, and a(-, •) is assumed to satisfy c0lMlm,2,O ) <ColMlm,2,0 for all u £ X. The solution of this problem exists according to the Lax-Milgram theorem. One way to solve this problem numerically is to approximate X by a sequence of finite-dimensional spaces Xn for A/eN. If {ui}i=XyN is a basis of XN , writing u in the form