474 RWTH Publication No: 712673        2018        IGPM474.pdf
TITLE Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints
AUTHORS Igor Voulis, Arnold Reusken
ABSTRACT We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that t 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.
KEYWORDS abstract parabolic problem, discontinuous Galerkin methods, discretization of linear constraints, optimal discretization error bounds
DOI 10.1515/jnma-2018-0013
PUBLICATION Journal of Numerical Mathematics
27(3), pp 155–182