A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids

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Abstract

In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651–1671, 2015). We first prove that the DG solution converges in the L2-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be p+ 2 , when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two (p+ 1) -degree right Radau polynomials in the x and y directions. The less significant part converges to zero at O(hp + 2). These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the L2-norm at O(hp + 2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.

Original languageEnglish (US)
Pages (from-to)945-974
Number of pages30
JournalJournal of Scientific Computing
Volume68
Issue number3
DOIs
StatePublished - Sep 1 2016

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A Posteriori Error Analysis
Cartesian Grid
Hyperbolic Conservation Laws
Discontinuous Galerkin Method
Galerkin methods
Error analysis
Conservation
Polynomials
Discontinuous Galerkin
Polynomial
Converge
Superconvergence
Norm
Tensor Product
Tensors
y direction
x direction
Mesh Refinement
A Posteriori Error Estimates
Order of Convergence

Keywords

  • A posteriori error estimates
  • Cartesian grids
  • Discontinuous Galerkin method
  • Hyperbolic problems
  • Superconvergence

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids",
abstract = "In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651–1671, 2015). We first prove that the DG solution converges in the L2-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be p+ 2 , when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two (p+ 1) -degree right Radau polynomials in the x and y directions. The less significant part converges to zero at O(hp + 2). These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the L2-norm at O(hp + 2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.",
keywords = "A posteriori error estimates, Cartesian grids, Discontinuous Galerkin method, Hyperbolic problems, Superconvergence",
author = "Mahboub Baccouch",
year = "2016",
month = "9",
day = "1",
doi = "10.1007/s10915-016-0166-0",
language = "English (US)",
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pages = "945--974",
journal = "Journal of Scientific Computing",
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PY - 2016/9/1

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N2 - In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651–1671, 2015). We first prove that the DG solution converges in the L2-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be p+ 2 , when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two (p+ 1) -degree right Radau polynomials in the x and y directions. The less significant part converges to zero at O(hp + 2). These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the L2-norm at O(hp + 2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.

AB - In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651–1671, 2015). We first prove that the DG solution converges in the L2-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be p+ 2 , when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two (p+ 1) -degree right Radau polynomials in the x and y directions. The less significant part converges to zero at O(hp + 2). These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the L2-norm at O(hp + 2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.

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