| Title:
|
A new finite element approach for problems containing small geometric details (English) |
| Author:
|
Hackbusch, W. |
| Author:
|
Sauter, S. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
105-117 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Finite Elements |
| Keyword:
|
Shortley-Weller discretization |
| Keyword:
|
complicated boundary |
| MSC:
|
65N15 |
| MSC:
|
65N30 |
| MSC:
|
65N50 |
| MSC:
|
65Y20 |
| MSC:
|
74A60 |
| MSC:
|
74M25 |
| MSC:
|
74S05 |
| idZBL:
|
Zbl 0912.65088 |
| idMR:
|
MR1629676 |
| . |
| Date available:
|
2009-02-17T10:10:36Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107637 |
| . |
| Reference:
|
[1] R. Bank, J. Xu. : An Algorithm for Coarsening Unstructured Meshes.Numer. Math., 73(1):1–36, 1996. Zbl 0857.65034, MR 1379277 |
| Reference:
|
[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 1423810 |
| Reference:
|
[3] W. Hackbusch. : On the Multi-Grid Method Applied to Difference Equations.Computing, 20:291–306, 1978. Zbl 0391.65045 |
| Reference:
|
[4] W. Hackbusch. : Multi-Grid Methods and Applications.Springer Verlag, 1985. Zbl 0595.65106 |
| Reference:
|
[5] W. Hackbusch. : Elliptic Differential Equations.Springer Verlag, 1992. Zbl 0875.35032, MR 1197118 |
| Reference:
|
[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 1444870 |
| Reference:
|
[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. |
| Reference:
|
[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. Zbl 0874.65086, MR 1431211 |
| Reference:
|
[9] R. Kornhuber, H. Yserentant. : Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid.Contemporay Mathematics, 180:49–60, 1994. Zbl 0817.65109, MR 1312377 |
| Reference:
|
[10] V. Mikulinsky. : Multigrid Treatment of Boundary and Free-Boundary Conditions.PhD thesis, The Weizmann Institute of Science, Rehovot 76100, Israel, 1992. |
| Reference:
|
[11] J. Ruge, K. Stüben. : Algebraic multigrid.In S. McCormick, editor, Multigrid Methods, pages 73–130, Pennsylvania, 1987. SIAM Philadelphia. MR 0972756 |
| Reference:
|
[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. |
| Reference:
|
[13] S. Sauter. : Vergröberung von Finite-Elemente-Räumen.Technical report, Universität Kiel, Germany, 1997. Habilitationsschrift. |
| Reference:
|
[14] G. H. Shortley, R. Weller. : Numerical Solution of Laplace’s Equation.J. Appl. Phys., 9:334–348, 1938. Zbl 0019.03801 |
| . |
