An optimal-order \(L^2\)-error estimate for nonsymmetric discontinuous Galerkin methods for a parabolic equation in multiple space dimensions. (English) Zbl 1227.65085
Summary: We analyze the nonsymmetric discontinuous Galerkin methods (NIPG and IIPG) for linear elliptic and parabolic equations with a spatially varied coefficient in multiple spatial dimensions. We consider \(d\)-linear approximation spaces on a uniform rectangular mesh, but our results can be extended to smoothly varying rectangular meshes. Using a blending or Boolean interpolation, we obtain a superconvergence error estimate in a discrete energy norm and an optimal-order error estimate in a semi-discrete norm for the parabolic equation. The \(L^{2}\)-optimality for the elliptic problem follows directly from the parabolic estimates. Numerical results are provided to validate our theoretical estimates. We also discuss the impact of penalty parameters on convergence behaviors of NIPG.
MSC:
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76R50 | Diffusion |
Keywords:
convergence analysis; discontinuous Galerkin methods; NIPG; IIPG; error estimates; superconvergenceReferences:
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