Convergence analysis of finite element method for a parabolic obstacle problem

Abstract

A conforming finite element method is proposed and analyzed for numerical approximation of the solution of a parabolic variational inequality of the obstacle problem. The model problem, which is originally proposed using a general obstacle, is reframed as a model problem with zero obstacle but with non-homogeneous Dirichlet boundary conditions. Subsequently the discrete problem is reframed and the error analysis proving the convergence of the method is performed. The analysis requires a positive preserving interpolation with non-homogeneous Dirichlet boundary condition and a post-processed solution that satisfies the boundary conditions sharply. The results in the article extend the results of (Johnson, SINUM, 1976) for a zero obstacle function to a more general obstacle function.