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A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

Published online by Cambridge University Press:  28 May 2015

Rongpei Zhang ,
Xijun Yu ,
Xia Cui ,
Xiaohan Long  and
Tao Feng
Rongpei Zhang*
Affiliation:
School of Sciences, Liaoning Shihua University, Fushun 113001, Liaoning, China
Xijun Yu*
Affiliation:
National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Xia Cui*
Affiliation:
National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Xiaohan Long*
Affiliation:
School of Mathematics and Information, Ludong University, Yantai 264025, Shandong, China
Tao Feng*
Affiliation:
National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is L2 stable. When the finite element space consists of interpolative polynomials of degrees k, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of . Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.

Keywords

65M6035K05Parabolic equationdiscontinuous coefficientdiscontinuous Galerkin methoderror estimatestability analysis
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 2 , May 2013 , pp. 325 - 342
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Yuang, G. W., Hang, X. D., Sheng, Z. Q. and Yue, J. Y., Progress in numerical methods for radiation diffusion equations, Chinese J. Comput. Phys., 26 (2009), pp. 475–500.Google Scholar
[2]Samarskii, A. A., Introduction to the Theory of Difference Schemes, Moscow, Nauka, 1971, in Russian.Google Scholar
[3]Shashkov, M. and Steinberg, S., Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys., 129 (1996), pp. 383–405.CrossRefGoogle Scholar
[4]Zhu, S. H., Yuang, G. W. and Sun, W. W., Convergence and stability of explicit/implicit schemes for parabolic equations with discontinuous coefficients, Int. J. Numer. Anal. Model., 1 (2004), pp. 131–146.Google Scholar
[5]Sinha, R. H. and Deka, B., Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal., 43 (2005), pp. 733–749.CrossRefGoogle Scholar
[6]Huang, H. X. and Li, Z. L., Convergence analysis of the immersed interface method, IMA J. Numer. Anal., 19 (1999), pp. 583–608.CrossRefGoogle Scholar
[7]Ewing, R. E., Li, Z. L., Lin, T. and Lin, Y. P., The immersed finite volume element methods for the elliptic interface problems, Math. Comput. Simulation, 50 (1999), pp. 63–76.CrossRefGoogle Scholar
[8]Li, Z., Lin, T., Lin, Y. and Rogers, R. C., An immersed finite element space and its approximation capability, Numer. Methods Part. Diff. Eq., 20 (2004), pp. 338–367.CrossRefGoogle Scholar
[9]Reed, W. H. and Hill, T. R., Triangular mesh methods for the neutron transport equation, Los Alamos Scienfic Laboratory Report, LA-UR-73-479, 1973.Google Scholar
[10]Cockburn, B. and Shu, C., TVB Runge-Kutta local projection discontinuous Galerkin finite element methods for conservation laws II: general framework, Math. Comput., 52 (1989), pp. 411–435.Google Scholar
[11]Cockburn, B., Lin, S. Y. and Shu, C., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84 (1989), pp. 90–113.CrossRefGoogle Scholar
[12]Cockburn, B. and Shu, C., TVB Runge-Kutta local projection discontinuous Galerkin finite element methods for conservation laws IV: the multidimensional case, Math. Comput.,54 (1990), pp. 545–581.Google Scholar
[13]Cockburn, B. and Shu, C., TVB Runge-Kutta local projection discontinuous Galerkin finite element methods for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199–224.CrossRefGoogle Scholar
[14]Cockburn, B. and Shu, C., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463.CrossRefGoogle Scholar
[15]Gassner, G., Lörcher, F. and Munz, C. A., A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, J. Comput. Phys., 224 (2007), pp. 1049–1063.Google Scholar
[16]Lörcher, F., Gassner, G. and Munz, C. A., An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations, J. Comput. Phys., 227 (2008), pp. 5649–5670.CrossRefGoogle Scholar
[17]Liu, H. L. and Yan, J., The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM J. Numer. Anal., 47 (2009), pp. 475–698.CrossRefGoogle Scholar
[18]Liu, H. L. and Yan, J., The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections, Commun. Comput. Phys., 3 (2010), pp. 541–564.Google Scholar
[19]Pietro, D. A. and Ern, A., Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions, Numer. Methods Part. Diff. Eq., Published online, DOI: 10.1002/num.20675.Google Scholar
[20]Ern, A., Stephansen, A. and Zunino, P, A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2009), pp. 235–256.CrossRefGoogle Scholar
[21]Ern, A. and Stephansen, A., A posteriori energy-norm error estimates for discontinuous Galerkin method with weighted averages for advection-diffusion equations approximated by weighted interiori penalty methods, J. Comput. Math., 26 (2008), pp. 488–510.Google Scholar
[22]Cai, Z. Q., Ye, X. and Zhang, S., Discontinuous Galerkin finite element methods for interface problems: apriori and a posteriori error estimations, SIAM J. Numer. Anal., to appear.Google Scholar
[23]Shu, C. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471.CrossRefGoogle Scholar