# FINITE ELEMENT MULTISCALE METHODS FOR POISSON’S EQUATION WITH RAPIDLY VARYING HETEROGENEOUS COEFFICIENTS

### Elfverson, D.; Malqvist, A.;

An abstract framework for constructing finite element multiscale methods is presented. Using this framework we propose and compare two different multiscale methods, one based on the continuous Galerkin finite element method and one on the discontinuous Galerkin finite element method. In these multiscale methods the solution is split into coarse and fine scale contributions. The fine scale contribution is obtained by solving localized constituent problems on patches and is used to obtain a modified coarse scale equation. The localized constituent problems are completely parallelizable i.e, no communication between the different problems are needed. The modified coarse scale equation has considerably less degrees of freedom than the original problem. Numerical experiments are presented where the effect of the patch size of the local constituent problems as well as the convergence of the multiscale methods are investigated and compared for the proposed multiscale methods. We conclude that for a given accuracy and a fixed number of patches, smaller patches can be used for the discontinuous Galerkin multiscale method compared to the continuous Galerkin multiscale method.

Palavras-chave: finite element methods, discontinuous Galerkin, multiscale methods,

Palavras-chave:

DOI: 10.5151/meceng-wccm2012-18132

##### Referências bibliográficas
• [1] J. Aarnes, B.-0. Hemsund, “Multiscale discontinuous Galerkin methods for elliptic problems with multiple scale”. Letc. notes in Comput. Sci. Eng. vol 44, Springer, Heidelberg, Berlin, 2005.
• [2] A. Abdulle, “Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scale”. Math. Comp. 81, 687-713, 201
• [3] D. N. Arnold, “An interior penalty finite element method with discontinuous elements”. SIAM J. Numer. Anal. 19, 742-760, 1982.
• [4] D. N. Arnold, F. Brezzi, B. Cockburn, L. Marini “Unified analysis of discontinuous Galerkin methods for elliptic problems”. SIAM J. Numer. Anal. 39, 1749-1779, 2001.
• [5] A. Babu?ska, J. E. Osborn, “Can a finite element method perform arbitrarily bad?”. Math. Comp. 69(230), 443-462, 2000.
• [6] G. A. Baker, “Finite element methods for elliptic equations using nonconforming elements”. Math. Comp 31, 45-59, 1977.
• [7] W. E, “Principles of multiscale modeling”. Math. Model. and Methods Cambridge university press, 2011.
• [8] Y. Efendiev, T. Y. Hou, “Multiscale finite element methods: Theory and Applications”. Surveys and Tut. in Appl. Math. Sci., vol 4, Springer, New York,2009
• [9] T. Y. Hou, X.-H. Wu, “A multiscale finite element method for elliptic problems in composite materials and porous media”. J. of Comput. Phys. 134, 169-189, 1997
• [10] T. Hughes, “Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods”. Comput. Methods Appl. Mech. Engrg. 166(1-2), 3-24, 1998.
• [11] M. G. Larson, A.M°alqvist, “Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems”. Comput. Mehtods Appl. Mech. Engrg. s 196(21-24), 2313-2324, 2007.
• [12] A. M°alqvist, “Multiscale methods for elliptic problems”. Multiscale Model. and Simul. 9, 1064-1086, 2011.
• [13] A. M°alqvist, D. Peterseim “Localization of elliptic multiscale problems”. arXiv:1110.0692. Submitted, 2011.
• [14] D. A. Di Pietro, A. Ern, “Mathematical aspect of discontinuous Galerkin methods”. Math´ematiques et Apllications vol 19, Springer, 2012.
• [15] B. Rivière, “Discontinuous Galerkin methods for solving elliptic and parabolic equations: Theory and Implementation”. Soc. for Industrial and Applied Math., Philadelphia, PA, USA. 2008
##### Como citar:

Elfverson, D.; Malqvist, A.; "FINITE ELEMENT MULTISCALE METHODS FOR POISSON’S EQUATION WITH RAPIDLY VARYING HETEROGENEOUS COEFFICIENTS", p. 836-845 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-18132

últimos 30 dias | último ano | desde a publicação