Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints

TL;DR

This paper develops and analyzes modified discontinuous Galerkin time discretization methods for parabolic problems with linear constraints, achieving optimal error bounds and superconvergence, improving upon standard methods.

Contribution

It introduces a modified DG method with projections for linear constraints, providing optimal error bounds and superconvergence results for constrained parabolic problems.

Findings

Modified DG method achieves optimal convergence rates.

Standard DG method exhibits sub-optimal convergence without modification.

Numerical experiments confirm theoretical error bounds.

Abstract

We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time dependent) Dirichlet boundary conditions and the time dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.