Finite element wavelets for solving partial differential equations

Abstract

This thesis deals with the application of wavelet bases for the numerical solution of operator equations, as boundary value problems and boundary integral equations. The use of suitable wavelet bases has the advantage that the arising stiffness matrices are well-conditioned uniformly in their sizes, allowing for an efficient iterative solution, and, for integral equations, that they allow for a compression of the in that case densely populated stiffness matrices to truly sparse ones without reducing the convergence rates. In this way, even for integral equations, a method of optimal computational complexity is obtained, in the sense that the work spent per degree of freedom is bounded, uniformly in their number.

One part of this thesis, viz. Chapters 3 and 4, is devoted to the construction of suitable wavelet bases on the domains, or manifolds, on which the equations to be solved are posed. Due to the generally nontrivial geometry of these domains, we can not rely on the classical construction of wavelets by means of shifts and dilates from one ‘mother’ wavelet. In Chapter 3, we follow another approach, introduced in [DS99], to construct wavelet bases on an important class of domains, viz. polytopes in IRn. Here, the idea is to construct wavelet bases for Lagrange finite element spaces. By a careful quantitative analysis of the statements from [DS99], we develop a theoretical framework which identifies in which way the available freedom in the wavelet construction can be used to obtain better conditioned finite element wavelet bases, yielding better conditioned stiffness matrices. The modified wavelets that we construct along these lines turn out to be relatively well-conditioned, where in comparison to the original construction, the condition numbers are up to a factor one thousand smaller.

A problem with this approach when constructing wavelets on manifolds is that, generally, it only yields locally supported wavelets when some modified L2 -inner product is used. When doing so, however, some of the obtained wavelets generally will not have vanishing moments, or more accurately, cancellation properties. What is more, generally the obtained wavelets cannot generate Riesz bases for Hs for s ≤ - ½. To tackle both problems, in Chapter 4 we follow an approach from [Ste04], that improves upon the construction from our paper [NS03], to construct well-localized ‘approximate wavelets’ that inherit all properties concerning cancellation properties and generation of Riesz bases for the range of Sobolev spaces obtained with truly L2 -biorthogonal space decompositions. Our numerical results demonstrate that these approximate wavelet bases are similarly conditioned as their counterparts on domains.

The other part of this thesis, viz. Chapters 2 and 5, deals with the application of wavelets for solving operator equations, where in particular we focus on matrix compression for integral equations. In Chapter 2, for general elliptic operator equations, we study which convergence rates can be expected for the Galerkin approximations. Based on the pioneering work [Sch98], and as a preparation for matrix compression, thinking of the stiffness matrix as being partitioned into blocks corresponding to the subdivision of the wavelet basis into levels, we derive precise conditions which errors can be allowed in the individual blocks without reducing the maximum possible rate. In Chapter 5, we study an important class of singular integral operators, containing the well-known operators resulting from the application of the boundary integral method. Based on two decay estimates for the entries of the stiffness matrix, and the aforementioned conditions from Chapter 2, we derive precise rules which entries have to be computed and which ones can be discarded without reducing the convergence rates. Partly because of our use of a new decay estimate from [Ste03], our dropping rules are simpler than and quantitatively improve upon those known from the literature. Our findings are numerically illustrated in a simple model example.