Error estimates for finite element methods for scalar conservation laws

Bernardo Cockburn; Pierre Alain Gremaud

doi:10.1137/0733028

Error estimates for finite element methods for scalar conservation laws

Bernardo Cockburn, Pierre Alain Gremaud

Mathematics

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

80 Scopus citations

Abstract

In this paper, new a posteriori error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least h^1/8 and h^1/4, respectively, in the L^∞(L¹)-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.

Original language	English (US)
Pages (from-to)	522-554
Number of pages	33
Journal	SIAM Journal on Numerical Analysis
Volume	33
Issue number	2
DOIs	https://doi.org/10.1137/0733028
State	Published - Apr 1996

Keywords

Discontinuous galerkin method
Error estimates
Multidimensional conservation laws
Streamline diffusion method

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abstract = "In this paper, new a posteriori error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least h1/8 and h1/4, respectively, in the L∞(L1)-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.",

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AU - Cockburn, Bernardo

AU - Gremaud, Pierre Alain

PY - 1996/4

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N2 - In this paper, new a posteriori error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least h1/8 and h1/4, respectively, in the L∞(L1)-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.

AB - In this paper, new a posteriori error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least h1/8 and h1/4, respectively, in the L∞(L1)-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.

KW - Discontinuous galerkin method

KW - Error estimates

KW - Multidimensional conservation laws

KW - Streamline diffusion method

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