2018/01/19 by Igor Voulis, Voulis, Igor, Arnold Reusken +1 · 1 citation
Engineering · Mathematics · #65J10 #65M60 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1801.06361
openalex publication_date 2018/01/19 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We consider time discretization methods for abstract parabolic problems with\ninhomogeneous linear constraints. Prototype examples that fit into the general\nframework are the heat equation with inhomogeneous (time dependent) Dirichlet\nboundary conditions and the time dependent Stokes equation with an\ninhomogeneous divergence constraint. Two common ways of treating such linear\nconstraints, namely explicit or implicit (via Lagrange multipliers) are\nstudied. These different treatments lead to different variational formulations\nof the parabolic problem. For these formulations we introduce a modification of\nthe standard discontinuous Galerkin (DG) time discretization method in which an\nappropriate projection is used in the discretization of the constraint. For\nthese discretizations (optimal) error bounds, including superconvergence\nresults, are derived. Discretization error bounds for the Lagrange multiplier\nare presented. Results of experiments confirm the theoretically predicted\noptimal convergence rates and show that without the modification the (standard)\nDG method has sub-optimal convergence behavior.\n