The dirichlet boundary conditions and related natural boundary conditions in strengthened sobolev spaces for discretized parabolic problems

Abstract

Correctness of initial boundary value problems and their discretizations are analyzed under unusual second-order boundary conditions, which can be considered as natural boundary conditions in strengthened Sobolev spaces and as improvements (in some cases) of the classical Dirichlet boundary conditions. Special attention is paid to optimal perturbation estimates for new variants of the penalty method with respect to the Dirichlet conditions.