Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem

We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in ℝd (d = 2, 3). For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete schemes, and derive error estimate of optimal order O(h+Δt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal O}(h + \Delta t)$$\end{document} in an energy norm. Moreover, the analysis is performed without any assumptions on the speed of propagation of the free boundary and only the realistic regularity ut∈ℒ2(0,T;ℒ2(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_t} \in {{\cal L}^2}(0,T;{{\cal L}^2}(\Omega ))$$\end{document} is assumed. Further, we present some numerical experiments to illustrate the performance of the proposed methods.