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A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

Published online by Cambridge University Press:  03 June 2015

Eric T. Chung  and
Wing Tat Leung
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Abstract

In this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.

Keywords

65M1265M60Multiscale problemsub-grid capturingmultiscale basis functionboundary layerinternal layer
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 2 , August 2013 , pp. 370 - 392
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Arnold, D. N.An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19 (1982), pp. 742–760.CrossRefGoogle Scholar
[2]Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D.Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779.CrossRefGoogle Scholar
[3]Brenner, S.Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal., 41 (2003), pp. 306–324.Google Scholar
[4]Castillo, P., Cockburn, B., Perugia, I. and Schoötzau, D.An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal., 38 (2000), pp. 1676–1706.Google Scholar
[5]Chung, E. and Efendiev, Y.Reduced contrast approximations for high-contrast multiscale flow problems. SIAM MMS, 8 (2010), pp. 1126–1153.Google Scholar
[6]Chung, E., Efendiev, Y. and Gibson, R.An energy-conserving discontinuous multiscale finite element method for the wave equation in heterogeneous media. Advances in Adaptive Data Analysis, 3 (2011), pp. 251–268.Google Scholar
[7]Chung, E. and Engquist, B.Optimal discontinuous galerkin methods for wave propagation. SIAM J. Numer. Anal., 44 (2006), pp. 2131–2158.Google Scholar
[8]Chung, E. and Engquist, B.Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal., 47 (2009), pp. 3820–3848.Google Scholar
[9]Chung, E. and Lee, C.A staggered discontinuous Galerkin method for the curl-curl operator. To appear in IMA J. Numer. Anal.Google Scholar
[10]Chung, E. and Lee, C.A staggered discontinuous Galerkin method for the convection-diffusion equation. J. Numer. Math., 20 (2012), pp. 1–31.CrossRefGoogle Scholar
[11]Cockburn, B. and Shu, C.-W.The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463.Google Scholar
[12]E, W. and Engquist, B.The heterogeneous multiscale methods. Commun. Math. Sci., 1 (2003), pp. 87–132.Google Scholar
[13]Efendiev, Y. and Hou, T. Y.Multiscale finite element methods, volume 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York, 2009.Google Scholar
[14]Efendiev, Y., Hou, T. and Wu, X.Convergence of a nonconformal multiscale finite element method. SIAM J. Numer. Anal., 37 (2000), pp. 888–910.Google Scholar
[15]Gibson, R., Gao, K., Chung, E. and Efendiev, Y.Multiscale modeling of acoustic wave propagation in two-dimensional media. Submitted.Google Scholar
[16]Hou, T. Y. and Liang, D.Multiscale analysis for convection dominated transport equations. DCDS-A, 23 (2009), pp. 281–298.Google Scholar
[17]Hou, T. and Wu, X.A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134 (1997), pp. 169–189.CrossRefGoogle Scholar
[18]Jenny, P., Lee, S.H. and Tchelepi, H.A.Multi-scale finite-volume method for elliptic problems in subsurface flow simulation J. Comput. Phys., 187 (2003), pp. 47–67.Google Scholar
[19]Kalashnikova, I., Farhat, C. and Tezaur, R.A discontinuous enrichment method for the finite element solution of high Péclet advection-diffusion problems. Finite elements in Analysis and Design, 45 (2009), pp. 238–250.Google Scholar
[20]Jiang, L. and Mishev, I.A mixed multiscale finite volume method for elliptic problems in two-phase flow simulations. Comm. Comput. Phys., 11 (2012), pp. 19–47.CrossRefGoogle Scholar
[21]Owhadi, H. and Zhang, L.Metric-based upscaling. Comm. Pure Appl. Math., 60 (2007), pp. 675–723.Google Scholar
[22]Park, P. and Hou, T. Y.Multiscale numerical methods for singularly-perturbed convection diffusion equations. International Journal of Computational Methods, 1 (2004), pp. 17–65.Google Scholar
[23]Riviere, B.Discontinuous Galerkin methods for solving elliptic and parabolic equations. SIAM, 2008.Google Scholar
[24]Wang, W., Guzman, J. and Shu, C.The multiscale discontinuous Galerkin method for solving a class of second order elliptic problems with rough coefficients. Int. J. Numer. Anal. Model., 8 (2011), pp. 28–47.Google Scholar