A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

  1. Clint Dawson,
  2. Jennifer Proft*

Article first published online: 4 OCT 2001

DOI: 10.1002/num.1026

Issue

Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations

Volume 17, Issue 6, pages 545–564, November 2001

Additional Information

How to Cite

Dawson, C. and Proft, J. (2001), A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations. Numerical Methods for Partial Differential Equations, 17: 545–564. doi: 10.1002/num.1026

Author Information

  1. Center for Subsurface Modeling—C0200, Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712

*Center for Subsurface Modeling—C0200, Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712

Publication History

  1. Issue published online: 4 OCT 2001
  2. Article first published online: 4 OCT 2001
  3. Manuscript Accepted: 1 MAR 2001
  4. Manuscript Received: 20 JUN 2000

Funded by

  • National Science Foundation. Grant Number: DMS-9805491

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Keywords:

  • discontinuous Galerkin method;
  • transport equation;
  • a priori error estimates

Abstract

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection-dominated convection-diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L∞(L2) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L2(L2) norm for the first penalty method with a penalty parameter of order one (h−1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001

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