Conforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem

Abstract

In this article, we propose and analyze conforming and discontinuous Galerkin (DG) finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in Rd(d=2,3). For fully discrete conforming method, we use globally continuous and piecewise linear finite element space. Whereas for the fully-discrete DG scheme, we employ piecewise linear finite element space for spatial discretization. The time discretization has been done by using the implicit backward Euler method. We present the error analysis for the conforming and the DG fully discrete schemes and derive an error estimate of optimal order O(h+Δt) in a certain energy norm defined precisely in the article. The analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that ut∈L2(0,T;L2(Ω)). The obstacle constraints are incorporated at the Lagrange nodes of the triangular mesh and the analysis exploits the Lagrange interpolation. We present some numerical experiment to illustrate the performance of the proposed methods.