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

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. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! " " " " " " " " " " " " " " " " " " " "