A PhD thesis @ NALAG

TW 2006_02

Jan Maes
Powell-Sabin spline based multiscale preconditioners for elliptic part ial differential equations

Advisor(s): Adhemar Bultheel

Abstract

In this dissertation we are concerned with the development of multilevel preconditioners for linear systems that arise from Galerkin methods for fourth order elliptic equations on two-dimensional polygonal domains. The key ingredients are the construction of multiscale bases for C1conforming finite element spaces of Powell-Sabin type, and the characterization of certain Sobolev spaces by weighted norm equivalences related to the multiscale representation of functions. The latter immediately yields bounds on the growth rate of the condition numbers of the preconditioned systems. Multiscale bases that characterize a large range of Sobolev spaces are preferable, since the corresponding preconditioners are more robust. We explore different types of multiscale bases, such as a suboptimal standard hierarchical basis, an optimal hierarchical basis based on Lagrange interpolation, and several wavelet-type bases. On non-uniform triangulations we construct a biorthogonal wavelet basis that characterizes the Sobolev spaces H^s(Ω) with s ∈ (0.802774, 2.5). On uniform triangulations we construct a semi-orthogonal wavelet basis that characterizes the Sobolev spaces H^s(Ω) with |s| < 2.5. Furthermore we develop an elegant way to extend the obtained results to similar constructions on the surface of the two-sphere.