Multilevel Preconditioning for Variational Problems

Abstract

AbstractFor the numerical solution of linear variational problems involving elliptic partial differential operators in n ≥ 2 space dimensions, iterative solution schemes are indispensable on account of their problem size. Our guiding principle is to devise iterative solvers which are optimal in the number of arithmetic operations, i.e., which are of linear complexity in the total number of unknowns. For these algorithms, asymptotically optimal preconditioners are required. The class of preconditioners for which this can be shown are of multilevel type, requiring nested approximation spaces to approximate the solution of the system on a fine user-specified grid. For smooth solutions of standard second and fourth order elliptic PDEs (partial differential equations) in variational form, approximations based on tensor products of higher-order B-splines yield high accuracy.For such problem classes, this survey collects the main ingredients for multilevel preconditioners in terms of higher order B-splines. There are three types of multilevel preconditioners for which asymptotic optimality can be shown. Two of them, the so-called additive preconditioners, are specified for isogeometric analysis involving linear elliptic partial differential operators in terms of variants of the BPX (Bramble-Pasciak-Xu) preconditioner and wavelet preconditioners. The third class are the so-called multiplicative preconditioners, specifically, multigrid methods.An essential ingredient for all these multilevel preconditioners are intergrid operators which transform vectors or matrices between grids of different grid spacing. For higher order B-splines, these intergrid operators can be derived from their refinement relations. In addition to a presentation of the theoretical ingredients, the performance of the different preconditioners will be demonstrated by some numerical examples.KeywordsVariational InequalityRiesz BasisBiorthogonal WaveletElliptic PDEsSpline SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.