A posteriori error analysis for elliptic pdes on domains with complicated structures

Abstract

The discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores. The model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh-width at most and mixed boundary conditions for the Poisson equation. Reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements. Emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates.

A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions

“…Then, for each Γ (i) N there exists a parallelogram ω i ⊂ Ω having Γ (i) N as one of its sides. It is proved in [5] that w 2…”

Adaptive discontinuous Galerkin methods for elliptic interface problems

“…Observing, however, that, upon interface approximation, the exact solution is defined on a different domain to its finite element approximation, the standard approach of proving a posteriori bounds, i.e., using PDE stability results linking the error with the residual, becomes cumbersome. Few a posteriori bounds for curved domains exist, focusing on the related (but simpler) problem of proving a posteriori error bounds for elliptic problems posed on one-compartment curved domains [21,2]; see also [20].…”

Functional a posteriori error estimates for problems with nonlinear boundary conditions

“…Estimate (4.23) follows from the trace theorem. More detailed information concerning such type inequalities and the constants can be found in the works of Sauter and Carstensen [2], S. G. Mikhlin [11] among others.…”