Algebraic Multilevel Preconditioning Methods, II

Abstract

Dedicated to the memory of Peter HenriciSummary. A recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed. It is based on a sequence of nested finite element spaces with the usual nodal basis functions. Using a nodeordering corresponding to the nested meshes, the finite element stiffness matrix is recursively split up into two-level block structures and is factored approximately in such a way that any successive Schur complement is replaced (approximated) by a matrix defined recursively and therefore only implicitely given. To solve a system with this matrix we need to perform a fixed number (v) of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level. It is shown that by a proper choice of iteration parameters it suffices to use v>(1-72) 2 iterations for the so constructed v-fold V-cycle (where v = 2 corresponds to a W-cycle) preconditioning matrices to be spectrally equivalent to the stiffness matrix. The conditions involve only the constant 7 in the strengthened C.-B.-S. inequality for the corresponding two-level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance. If we use successive uniform refinements of the meshes the method is of optimal order of computational complexity, if 72< w Under reasonable assumptions of the finite element mesh, the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method, for instance.

BPX-preconditioning for isogeometric analysis

“…We call this an (asymptotically) optimal preconditioner. The type of schemes for which this can be shown are the so-called additive preconditioners like the wavelet preconditioner and the BPX-preconditioner [9] whose optimality was proved independently in [10,20], multiplicative versions like multigrid [7,13], and algebraic multilevel iteration (AMLI) methods [1,2]. Multigrid preconditioners for isogeometric analysis have been analysed in [12], whereas domain decomposition type preconditioners have been proposed in [4,5,16].…”

“…Here, H r 0 (Ω) denotes the subset of H r (Ω) with homogenous essential boundary conditions. Essential, for a favorable performance of the discretization method for (1), is a fast solution scheme for the final, large, linear system which has to be solved. In view of its size and the sparsity and structure of the system matrix, one typically employs an iterative solver for the resulting linear system of equations whose convergence speed depends on the spectral condition number κ 2 (A) of the system matrix A.…”

A new robust multigrid method for 2D convection-diffusion problems

“…For problems where the diffusion is dominant such a result is expected e.g. from the analysis in [1], [3], [6J and mathematically founded by the strengthened Cauchy inequality. It appears that the preconditioning of SA by S:t is also good for many strongly nonsymmetric problems.…”

“…By nh we denote the "fine" mesh that results after a standard uniform refinement of nH. [3], [1], [5]) with a constant I < 1, which can be formulated in this setting as…”

“…For example, polynomial multilevel preconditioners based on incomplete block factorizations are treated in [1].…”