Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

Abstract

Abstract. We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form h α u W s,p (Ω) for positive and negative s and α, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N , the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results-previously known only for quasi-uniform meshes-to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

Finite elements on degenerate meshes: inverse-type inequalities and applications

“…The estimate (3.7) follows from the proof of Theorem 4.2 in Dahmen et al (2004) in the case when τ and τ are planar triangular elements. In this paper we allow the more general setting where τ and τ can be curved surface elements and the proof in (Dahmen et al, 2004) needs to be extended slightly.…”

“…Such an extension has been obtained in Dahmen et al (2004) for shape-regular meshes at the expense of working in Besov norms. We have avoided such extensions here in order to simplify the present paper.…”

“…In this paper we allow the more general setting where τ and τ can be curved surface elements and the proof in (Dahmen et al, 2004) needs to be extended slightly.…”

“…However, practical meshes are often non-quasiuniform and then the negative powers of h min in (1.1) may give rise to overly pessimistic convergence rates. In Dahmen et al (2004), less pessimistic 380 I.G. GRAHAM ET AL.…”

“…to the analysis of quadrature errors in discrete Galerkin boundary element methods (Graham et al, 2000a) and to the analysis of the mortar element method (Dahmen et al, 2004). In fact, Dahmen et al (2004) contains more general versions of (1.2), e.g. in the Sobolev space W s, p (Ω ) and in related Besov spaces.…”