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

Abstract

In this paper we obtain a range of inverse-type inequalities which are applicable to finite-element functions on general classes of meshes, including degenerate meshes obtained by anisotropic refinement. These are obtained for Sobolev norms of positive, zero and negative order. In contrast to classical inverse estimates, negative powers of the minimum mesh diameter are avoided. We give two applications of these estimates in the context of boundary elements: (i) to the analysis of quadrature error in discrete Galerkin methods and (ii) to the analysis of the panel clustering algorithm. Our results show that degeneracy in the meshes yields no degradation in the approximation properties of these methods.

Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems

“…Using the E -elementwise best approximation property of Π Γ and the local inverse estimate from [22], Theorem 3.6, we obtain…”

Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods

“…With this and standard scaling arguments, one obtains µ 2 26) where the second estimate holds because of norm equivalence on finite dimensional spaces. By use of (4.26) with µ = (1 − Π)ψ, we conclude…”

“…[13, Theorem 4.1]) as well as the inverse estimate from [26,Theorem 3.6]. The main task now is to bound the last term of the preceding estimate appropriately and to absorb it on the left-hand side.…”

“…for all k ∈ N. Here, the constant C > 0 stems from the inverse estimate in [26,Theorem 3.6] and is independent of , k ∈ N. Consequently, we may estimate…”

“…where C > 0 again stems from the inverse estimate in [26,Theorem 3.6]. With h 0 > 0 from Proposition 4.2, we choose…”