Geodesic
A new finite element approach for problems containing small geometric details
Archivum mathematicum, Tome 34 (1998) no. 1, pp. 105-117 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper a new finite element approach is presented which allows the discretization of PDEs on domains containing small micro-structures with extremely few degrees of freedom. The applications of these so-called Composite Finite Elements are two-fold. They allow the efficient use of multi-grid methods to problems on complicated domains where, otherwise, it is not possible to obtain very coarse discretizations with standard finite elements. Furthermore, they provide a tool for discrete homogenization of PDEs without requiring periodicity of the data.
In this paper a new finite element approach is presented which allows the discretization of PDEs on domains containing small micro-structures with extremely few degrees of freedom. The applications of these so-called Composite Finite Elements are two-fold. They allow the efficient use of multi-grid methods to problems on complicated domains where, otherwise, it is not possible to obtain very coarse discretizations with standard finite elements. Furthermore, they provide a tool for discrete homogenization of PDEs without requiring periodicity of the data.
Classification : 65N15, 65N30, 65N50, 65Y20, 74A60, 74M25, 74S05
Keywords: Finite Elements; Shortley-Weller discretization; complicated boundary 
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     title = {A new finite element approach for problems containing small geometric details},
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     url = {http://geodesic.mathdoc.fr/item/ARM_1998_34_1_a10/}
}
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Hackbusch, W.; Sauter, S. A new finite element approach for problems containing small geometric details. Archivum mathematicum, Tome 34 (1998) no. 1, pp. 105-117. http://geodesic.mathdoc.fr/item/ARM_1998_34_1_a10/

[1] R. Bank, J. Xu. : An Algorithm for Coarsening Unstructured Meshes. Numer. Math., 73(1):1–36, 1996. | MR | Zbl

[2] R. E. Bank, J. Xu. : A Hierarchical Basis Multi-Grid Method for Unstructured Grids. In W. Hackbusch and G. Wittum, editors, Fast Solvers for Flow Problems, Proceedings of the Tenth GAMM-Seminar, Kiel. Verlag Vieweg, 1995. | MR

[3] W. Hackbusch. : On the Multi-Grid Method Applied to Difference Equations. Computing, 20:291–306, 1978. | Zbl

[4] W. Hackbusch. : Multi-Grid Methods and Applications. Springer Verlag, 1985. | Zbl

[5] W. Hackbusch. : Elliptic Differential Equations. Springer Verlag, 1992. | MR | Zbl

[6] W. Hackbusch, S. Sauter. : Adaptive Composite Finite Elements for the Solution of PDEs Containing non-uniformly distributed Micro-Scales. Matematicheskoe Modelirovanie, 8(9):31–43, 1996. | MR

[7] W. Hackbusch, S. Sauter. : Composite Finite Elements for Problems Containing Small Geometric Details. Part II: Implementation and Numerical Results. Computing and Visualization in Science, 1(1):15–25, 1997.

[8] W. Hackbusch, S. Sauter. : Composite Finite Elements for the Approximation of PDEs on Domains with Complicated Micro-Structures. Numer. Math., 75(4):447–472, 1997. | MR | Zbl

[9] R. Kornhuber, H. Yserentant. : Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid. Contemporay Mathematics, 180:49–60, 1994. | MR | Zbl

[10] V. Mikulinsky. : Multigrid Treatment of Boundary and Free-Boundary Conditions. PhD thesis, The Weizmann Institute of Science, Rehovot 76100, Israel, 1992.

[11] J. Ruge, K. Stüben. : Algebraic multigrid. In S. McCormick, editor, Multigrid Methods, pages 73–130, Pennsylvania, 1987. SIAM Philadelphia. | MR

[12] S. Sauter. : Composite finite elements for problems with complicated boundary. Part III: Essential boundary conditions. Technical report, Lehrstuhl Praktische Mathematik, Universität Kiel, 1997. submitted to Computing and Visualization in Sciences.

[13] S. Sauter. : Vergröberung von Finite-Elemente-Räumen. Technical report, Universität Kiel, Germany, 1997. Habilitationsschrift.

[14] G. H. Shortley, R. Weller. : Numerical Solution of Laplace’s Equation. J. Appl. Phys., 9:334–348, 1938. | Zbl