| Citation: |
Runchang Lin, Xiu Ye, Shangyou Zhang, Peng Zhu. ANALYSIS OF A DG METHOD FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 830-841. doi: 10.11948/20180164
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ANALYSIS OF A DG METHOD FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS
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Abstract
In this article, we studied a discontinuous Galerkin finite element method for convection-diffusion-reaction problems with singular perturbation. Our approach is highly flexible by allowing the use of discontinuous approximating function on polytopal mesh without imposing extra conditions on the convection coefficient. A priori error estimate is devised in a suitable energy norm on general polytopal mesh. Numerical examples are provided. -
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References
[1] P. F. Antonietti, P. Houston, X. Hu, et al, Multigrid algorithms for hp-version interiori penalty discontinuous Galerkin methods on polygonal and polyhedral meshes, Calcolo, 2017, 54, 1169-1198. doi: 10.1007/s10092-017-0223-6 [2] B. Ayuso and L. D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 2009, 47(2), 1391-1420. [3] C. E. Baumann and J. T. Oden, A discontinuous $hp$ finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 1999, 175(3-4), 311-341. [4] A. Buffa, T. J. R. Hughes, and G. Sangalli, Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems, SIAM J. Numer. Anal., 2006, 44, 1420-1440. doi: 10.1137/050640382 [5] A. Cangiani, E. H. Georgoulis, T. Pryer and O. J. Sutton, A posteriori error estimates for the virtual element method, Numer. Math., 2017, 137, 857-893. doi: 10.1007/s00211-017-0891-9 [6] J. Du and E. Chung, An adaptive staggered discontinuous Galerkin method for steady steady state convection-diffusion equation, J. Sci. Comput., 2018. DOI: 10.1007/s10915-018-0695-9. [7] P. Houston, C. Schwab, and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 2002, 39(6), 2133-2163. doi: 10.1137/S0036142900374111 [8] T. Linß, Layer-adapted meshes for reaction-convection-diffusion problems, Springer-Verlag, Berlin, 2010. [9] J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [10] D. A. Di Pietro and A. Ern, Hybrid high-order methods for variable-diffusion problems on general meshes, C.R. Math. Acad. Sci. Paris, 2015, 353(1), 31-34. doi: 10.1016/j.crma.2014.10.013 [11] H.-G. Roos, M. Stynes, and L. Tobiska, Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems(Second edition), Springer-Verlag, Berlin, 2008. [12] C. Talischi, A family of $H(div)$ finite element approximations on polygonal meshes, SIAM J. Sci. Comput., 2015, 37, 1067-1088. doi: 10.1137/140979873 [13] J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 2014, 83, 2101-2126. doi: 10.1090/S0025-5718-2014-02852-4 -
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Runchang Lin, Xiu Ye, Shangyou Zhang, Peng Zhu. ANALYSIS OF A DG METHOD FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 830-841. doi: 10.11948/20180164
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Figure 1. The uniform rectangular meshes of
$ h = 1/2, 1/4 $ , and the uniform pentagonal meshes of$ h = 1/2, 1/4 $ . -
Figure 2. Example 4.2. The
$ {\mathbb P}_1 $ numerical solution for$ \epsilon = 10^{-4} $ (top) and$ \epsilon = 10^{-10} $ , on a$ 64\times 64 $ rectangle mesh. -
Figure 3. Example 4.3. The
$ P_2 $ and$ P_4 $ numerical solutions for$ \epsilon = 10^{-9} $ over a mesh of 1, 024 elements. -
Figure 4. Example 4. The numerical
$ {\mathbb P}_1 $ and$ {\mathbb P}_4 $ solutions for$ \epsilon = 10^{-9} $ over a mesh of 1, 024 elements.