REDUCED MULTISCALE COMPUTATION ON ADAPTED GRID FOR THE CONVECTION-DIFFUSION ROBIN PROBLEM

Shan Jiang; Meiling Sun; Yin Yang

doi:10.11948/2017091

2017 Volume 7 Issue 4

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Shan Jiang, Meiling Sun, Yin Yang. REDUCED MULTISCALE COMPUTATION ON ADAPTED GRID FOR THE CONVECTION-DIFFUSION ROBIN PROBLEM[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1488-1502. doi: 10.11948/2017091

Citation:

Shan Jiang, Meiling Sun, Yin Yang. REDUCED MULTISCALE COMPUTATION ON ADAPTED GRID FOR THE CONVECTION-DIFFUSION ROBIN PROBLEM[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1488-1502. doi: 10.11948/2017091

REDUCED MULTISCALE COMPUTATION ON ADAPTED GRID FOR THE CONVECTION-DIFFUSION ROBIN PROBLEM

1 School of Science, Nantong University, Nantong 226019, China;
2 Department of Mathematics, Institute for Scientific Computation, Texas A & M University, College Station, TX 77843, USA;
3 Department of Public Courses, Nantong Vocational University, Nantong 226007, China;
4 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan 411105, China

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Abstract

Abstract

We propose a reduced multiscale finite element method for a convectiondiffusion problem with a Robin boundary condition. The small perturbed parameter would cause boundary layer oscillations, so we apply several adapted grids to recover this defect. For a Robin boundary relating to derivatives, special interpolating strategies are presented for effective approximation in the FEM and MsFEM schemes, respectively. In the multiscale computation, the multiscale basis functions can capture the local boundary layer oscillation, and with the help of the reduced mapping matrix we may acquire better accuracy and stability with a less computational cost. Numerical experiments are provided to show the convergence and efficiency.
- Multiscale finite element computation /
- Singular perturbation /
- Robin problem /
- Adapted grid /
- Reduced matrix
MSC: 35J25;65N12;65N30

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References

[1]	A. Abdulle and Y. Bai, Adaptive reduced basis finite element heterogeneous multiscale method, Comput. Methods Appl. Mech. Engrg., 2013, 257, 203-220. Google Scholar
[2]	X. Cai, D. L. Cai, R. Q. Wu and K. H. Xie, High accuracy non-equidistant method for singular perturbation reaction-diffusion problem, Appl. Math. Mech., 2009, 30(2), 175-182. Google Scholar
[3]	E. T. Chung, Y. Efendiev and W. T. Leung, Residual-driven online generalized multiscale finite element methods, J. Comput. Phys., 2015, 302, 176-190. Google Scholar
[4]	W. N. E, B. Engquist, X. T. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods:A review, Commun. Comput. Phys., 2007, 2(3), 367-450. Google Scholar
[5]	Y. Efendiev, J. Galvis and T. Y. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys., 2013, 251, 116-135. Google Scholar
[6]	Y. Efendiev, R. Lazarov, M. Moon and K. Shi, A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems, Comput. Methods Appl. Mech. Engrg., 2015, 292, 243-256. Google Scholar
[7]	P. Henning, M. Ohlberger and B. Schweizer, An adaptive multiscale finite element method, Multiscale Model. Simul., 2014, 12, 1078-1107. Google Scholar
[8]	T. Y. Hou and P. F. Liu, Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients, Discrete Contin. Dynam. Sys. A, 2016, 36(8), 4451-4476. Google Scholar
[9]	Y. Q. Huang, K. Jiang and N. Y. Yi, Some weighted averaging methods for gradient recovery, Adv. Appl. Math. Mech., 2012, 4(2), 131-155. Google Scholar
[10]	Y. Q. Huang, Y. F. Su, H. Y. Wei and N. Y. Yi, Anisotropic mesh generation methods based on ACVT and natural metric for anisotropic elliptic equation, Science China:Mathematics, 2013, 56(12), 2615-2630. Google Scholar
[11]	L. J. Jiang and Q. Q. Li, Reduced multiscale finite element basis methods for elliptic PDEs with parameterized inputs, J. Comput. Appl. Math., 2016, 301, 101-120. Google Scholar
[12]	S. Jiang and Y. Q. Huang, Numerical investigation on the boundary conditions for the multiscale base functions, Commun. Comput. Phys., 2009, 5(5), 928-941. Google Scholar
[13]	S. Jiang, M. Presho and Y. Q. Huang, An adapted Petrov-Galerkin multiscale finite element for singularly perturbed reaction-diffusion problems, Inter. J. Comput. Math., 2016, 93(7), 1200-1211. Google Scholar
[14]	J. Kevorkian, and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York, 1996. Google Scholar
[15]	R. C. Lin and M. Stynes, A balanced finite element method for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal., 2012, 50(5), 2729-2743. Google Scholar
[16]	N. Madden and S. Russell, A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction-diffusion problem, Adv. Comput. Math., 2015, 41, 987-1014. Google Scholar
[17]	J. J. Miller, E. O'Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems (revised edition), World Scientific, Singapore, 2012. Google Scholar
[18]	A. Muntean and M. N. Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media, J. Math. Anal. Appl., 2010, 371(2), 705-718. Google Scholar
[19]	M. Presho and J. Galvis, A mass conservative generalized multiscale finite element method applied to two-phase flow in heterogeneous porous media, J. Comput. Appl. Math., 2016, 296, 376-388. Google Scholar
[20]	K. Sharma, P. Rai and K. C. Patidar, A review on singularly perturbed differential equations with turning points and interior layers, Appl. Math. Comput., 2013, 219(22), 10575-10609. Google Scholar
[21]	G. I. Shishkin, A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation, Numer. Math. Theor. Meth. Appl., 2008, 1(2), 214-234. Google Scholar
[22]	F. Song, W. B. Deng and H. J. Wu, A combined finite element and oversampling multiscale Petrov-Galerkin method for the multiscale elliptic problems with singularities, J. Comput. Phys., 2016, 305, 722-743. Google Scholar
[23]	M. L. Sun and S. Jiang, Multiscale basis functions for singular perturbation on adaptively graded meshes, Advan. Appl. Math. Mech., 2014, 6(5), 604-614. Google Scholar