Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements

Abstract

This paper analyzes two-level Schwarz methods for matrices arising from the p-version finite element method on triangular and tetrahedral meshes. The coarse level consists of the lowest order finite element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a model problem.

A high-order discontinuous Galerkin method for the unsteady incompressible Navier–Stokes equations

“…Second, efficient preconditioning for the Poisson pressure equation should be devised. The class of optimal two-level overlapping Schwarz preconditioners for the p-version finite element methods recently introduced in [35] can be applicable in the discontinuous Galerkin setting.…”

Preconditioners for the indefinite linear system arising from the hp discretization of Maxwell's equations

“…In addition to the new sets of basis functions, Zaglmayr [12] also proposed an efficient preconditioner for solving linear systems resulting from the discretization of the weak form of (1) for problems where >0. Such a preconditioner is a generalization of the one proposed by Schöberl et al [14] for H 1 -conforming problems. The purpose of this work is to examine how to construct a preconditioner for the indefinite linear equation system that is obtained when a higher-order H(curl) discretization of (1) is adopted for the case where <0.…”

A stabilized finite element method for advection-diffusion equations on surfaces

“…There is a big literature, in particular high order methods and three dimensional problems are treated in [2,5,7,8,9,18,22,26,27,29,29,34,35,36,38,40]. There is a classical paper on multi-level analysis for h-version DG methods by Gopalakrishnan and Kanschat [17], and a recent one studying higher order methods by Antonietti and Houston [3] showing a polynomial growth of the condition number in p. We will see that the conditioning is significantly improved by hybridization, namely to a poly-logarithmic growth.…”