Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Abstract

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the BognerFox-Schmit rectangular element and the product two-point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners.

A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon

“…Our analysis requires higher than optimal solution regularity assumptions which are typical for OSC schemes. The linear system of the quadrature Galerkin scheme can efficiently be solved by a multilevel method developed in [5]. In this article, we also demonstrate that the standard analysis of the quadrature finite element Galerkin scheme based on the First Strang Lemma yields a subobtimal order error estimate, whereas our analysis based on an equivalent OSC problem gives optimal order error estimates.…”

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

“…Such extra regularity assumption on the exact solution is essential for analysis of C 1 spline collocation methods that do not involve integrals, but require evaluation of functions at certain quadrature points [7,10,19]. The analysis in [1,2] for finite element Galerkin method with quadrature for a linear biharmonic problem, restricted to the C 1 cubic spline (r = 3) case, requires H r+5 ( ) regularity. As demonstrated in [6,Section 5], the extra regularity assumptions on the solution and coefficients may be required only in quadrature finite element analysis, and may not affect convergence rates in practical computations.…”

“…It is common in quadrature finite element analysis to assume extra regularity [1,2,6,7,10,19]. We require H r+3 ( ) regularity, mainly due to technical details involved in analysis of the quadrature error in Lemma 3.1.…”

“…The finite element Galerkin method for biharmonic problems based on a weak formulation requires an H 2 trial space, see the recent work [1,2] and references therein. A byproduct of the analysis in this work in Section 3 yields optimal order convergence of a quadrature finite element Galerkin solution for a nonlinear biharmonic problem, extending the analysis in [2].…”

“…The major part of the proof is then to bound each of the eight terms.Proof of Theorem 4 1. We first consider the first two terms of the method in (1.15) for n = 1, .…”