Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints
This work provides a rigorous numerical analysis for treating linear constraints in parabolic problems, offering optimal error bounds that are important for computational scientists working on time-dependent PDEs with constraints.
The paper develops modified discontinuous Galerkin time discretization methods for parabolic problems with linear constraints, such as the heat equation with Dirichlet boundary conditions and the Stokes equation. The modified methods achieve optimal convergence rates, including superconvergence, whereas standard DG methods exhibit sub-optimal convergence.
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.