Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints

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...

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Bibliographic Details
Published in:arXiv.org 2018-06
Main Authors: Voulis, Igor, Reusken, Arnold
Format: Article
Language:English
Subjects:
Boundary conditions
Convergence
Dirichlet problem
Discretization
Divergence
Forecasting
Formulations
Galerkin method
Lagrange multiplier
Thermodynamics
Time dependence
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Summary: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.
ISSN:2331-8422