An optimal-order L2-error estimate for nonsymmetric discontinuous Galerkin methods for a parabolic equation in multiple space dimensions

Kaixin Wang; Hong Wang; Shuyu Sun; Mary F. Wheeler

doi:10.1016/j.cma.2009.02.006

An optimal-order L²-error estimate for nonsymmetric discontinuous Galerkin methods for a parabolic equation in multiple space dimensions

Kaixin Wang, Hong Wang, Shuyu Sun^*, Mary F. Wheeler

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

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 L²-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.

Original language	English (US)
Pages (from-to)	2190-2197
Number of pages	8
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	198
Issue number	27-29
DOIs	https://doi.org/10.1016/j.cma.2009.02.006
State	Published - May 15 2009
Externally published	Yes

Keywords

Convergence analysis
Discontinuous Galerkin methods
Error estimates
IIPG
NIPG
Superconvergence

ASJC Scopus subject areas

Computational Mechanics
Mechanics of Materials
Mechanical Engineering
General Physics and Astronomy
Computer Science Applications

Access to Document

10.1016/j.cma.2009.02.006

Cite this

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title = "An optimal-order L2-error estimate for nonsymmetric discontinuous Galerkin methods for a parabolic equation in multiple space dimensions",

abstract = "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 L2-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.",

keywords = "Convergence analysis, Discontinuous Galerkin methods, Error estimates, IIPG, NIPG, Superconvergence",

author = "Kaixin Wang and Hong Wang and Shuyu Sun and Wheeler, {Mary F.}",

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T1 - An optimal-order L2-error estimate for nonsymmetric discontinuous Galerkin methods for a parabolic equation in multiple space dimensions

AU - Wang, Kaixin

AU - Wang, Hong

AU - Sun, Shuyu

AU - Wheeler, Mary F.

PY - 2009/5/15

Y1 - 2009/5/15

N2 - 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 L2-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.

AB - 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 L2-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.

KW - Convergence analysis

KW - Discontinuous Galerkin methods

KW - Error estimates

KW - IIPG

KW - NIPG

KW - Superconvergence

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