| Title:
|
Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods (English) |
| Author:
|
Hoppe, Ronald H.W. |
| Author:
|
Wohlmuth, Barbara |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
40 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
227-248 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a posteriori error estimators which can be derived by a defect correction in higher order ansatz spaces or by taking advantage of superconvergence results. The performance of the algorithms is illustrated by several numerical examples. (English) |
| Keyword:
|
elliptic boundary value problems |
| Keyword:
|
mixed finite element methods |
| Keyword:
|
adaptive multilevel techniques |
| MSC:
|
35J25 |
| MSC:
|
65F10 |
| MSC:
|
65N12 |
| MSC:
|
65N15 |
| MSC:
|
65N30 |
| MSC:
|
65N50 |
| MSC:
|
65N55 |
| idZBL:
|
Zbl 0833.65131 |
| idMR:
|
MR1332315 |
| DOI:
|
10.21136/AM.1995.134292 |
| . |
| Date available:
|
2009-09-22T17:48:01Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134292 |
| . |
| Reference:
|
[LIT1] D.N. Arnold, F. Brezzi: Mixed and nonconforming finite element methods: implementation, post-processing and error estimates.$ M^2AN $ Math. Modelling Numer. Anal. 19, 7–35 (1985). MR 0813687, 10.1051/m2an/1985190100071 |
| Reference:
|
[LIT2] I. Babuska, W.C. Rheinboldt: Error estimates for adaptive finite element computations.SIAM J. Numer. Anal. 15, 736–754 (1978). MR 0483395, 10.1137/0715049 |
| Reference:
|
[LIT3] I. Babuska, W.C. Rheinboldt: A posteriori error estimates for the finite element method.Int. J. Numer. Methods Eng. 12, 1597–1615 (1978). 10.1002/nme.1620121010 |
| Reference:
|
[LIT4] R.E. Bank: PLTMG—A Software Package for Solving Elliptic Partial Differential Equations. User’s Guide 6.0..SIAM, Philadelphia, 1990. MR 1052151 |
| Reference:
|
[LIT5] R.E. Bank, A.H. Sherman, A. Weiser: Refinement algorithm and data structures for regular local mesh refinement.Scientific Computing, R. Stepleman et al. (eds.), IMACS North-Holland, Amsterdam, 1983, pp. 3–17. MR 0751598 |
| Reference:
|
[LIT6] R.E. Bank, A. Weiser: Some a posteriori error estimators for elliptic partial differential equations.Math. Comp. 44, 283–301 (1985). MR 0777265, 10.1090/S0025-5718-1985-0777265-X |
| Reference:
|
[LIT8] F. Bornemann, B. Erdmann, Kornhuber: A posteriori error estimates for elliptic problems in two and three space dimensions.Konrad-Zuse-Zentrum für Informationstechnik Berlin. Preprint SC 93–29, 1993. |
| Reference:
|
[LIT7] F. Bornemann, H. Yserentant: A basic norm equivalence for the theory of multilevel methods.Numer. Math. 64, 445–476 (1993). MR 1213412, 10.1007/BF01388699 |
| Reference:
|
[LIT9] D. Braess, R. Verfürth: A posteriori error estimators for the Raviart-Thomas element.Ruhr-Universität Bochum, Fakultät für Mathematik, Bericht Nr. 175, 1994. |
| Reference:
|
[LIT10] S. Brenner: A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method.SIAM J. Numer. Anal. 29, 647–678 (1992). Zbl 0759.65080, MR 1163350, 10.1137/0729042 |
| Reference:
|
[LIT11] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods.Springer, Berlin-Heidelberg-New York, 1991. MR 1115205 |
| Reference:
|
[LIT12] Z. Cai, C.I. Goldstein, J. Pasciak: Multilevel iteration for mixed finite element systems with penalty.SIAM J. Sci. Comput. 14, 1072–1088 (1993). MR 1232176, 10.1137/0914065 |
| Reference:
|
[LIT13] L.C. Cowsar: Domain decomposition methods for nonconforming finite element spaces of Lagrange-type.Rice University, Houston. Preprint TR 93–11, 1993. |
| Reference:
|
[LIT14] P. Deuflhard, P. Leinen, H. Yserentant: Concepts of an adaptive hierarchical finite element code.IMPACT Comput. Sci. Engrg. 1, 3–35 (1989). 10.1016/0899-8248(89)90018-9 |
| Reference:
|
[LIT15] R.E. Ewing, J. Wang: The Schwarz algorithm and multilevel decomposition iterative techniques for mixed finite element methods.Proc. 5th Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, D.F. Keyes et al. (eds.), SIAM, Philadelphia, 1992, pp. 48–55. MR 1189562 |
| Reference:
|
[LIT16] R.E. Ewing, J. Wang: Analysis of the Schwarz algorithm for mixed finite element methods.$ M^2AN $ Math. Modelling and Numer. Anal. 26, 739–756 (1992). MR 1183415, 10.1051/m2an/1992260607391 |
| Reference:
|
[LIT17] R.E. Ewing, J. Wang: Analysis of multilevel decomposition iterative methods for mixed finite element methods.$ M^2AN $ Math. Modelling and Numer. Anal. 28, 377–398 (1994). MR 1288504, 10.1051/m2an/1994280403771 |
| Reference:
|
[LIT18] B. Fraeijs de Veubeke: Displacement and equilibrium models in the finite element method.Stress Analysis, C. Zienkiewicz and G. Holister (eds.), John Wiley and Sons, New York, 1965. |
| Reference:
|
[LIT19] R.H.W. Hoppe, B. Wohlmuth: Element-oriented and edge-oriented local error estimators for nonconforming finite elements methods..Submitted to $ M^2 AN $ Math. Modelling and Numer. Anal. |
| Reference:
|
[LIT20] R.H.W. Hoppe, B. Wohlmuth: Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems.Submitted to SIAM J. Numer. Anal. MR 1461801 |
| Reference:
|
[LIT21] R.H.W. Hoppe, B. Wohlmuth: Adaptive iterative solution of mixed finite element discretizations using multilevel subspace decompositions and a flux-oriented error estimator.In preparation. |
| Reference:
|
[LIT22] P. Oswald: On a BPX-preconditioner for P1 elements.Computing 51, 125–133 (1993). Zbl 0787.65018, MR 1248895, 10.1007/BF02243847 |
| Reference:
|
[LIT23] J. Roberts, J.M. Thomas: Mixed and hybrid methods.Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions (eds.), Vol.II, Finite Element Methods (Part 1), North-Holland, Amsterdam, 1989. MR 1115239 |
| Reference:
|
[LIT24] P.S. Vassilevski, J. Wang: Multilevel iterative methods for mixed finite elements discretizations of elliptic problems.Numer. Math. 63, 503–520 (1992). MR 1189534, 10.1007/BF01385872 |
| Reference:
|
[LIT25] R. Verfürth: A review of a posteriori error estimation and adaptive mesh-refinement techniques.Manuscript, 1993. |
| Reference:
|
[LIT26] B. Wohlmuth, R.H.W. Hoppe: Multilevel approaches to nonconforming finite elements discretizations of linear second order elliptic boundary value problems.To appear in Journal of Computation and Information. |
| Reference:
|
[LIT27] J. Xu: Iterative methods by space decomposition and subspace correction.SIAM Rev. 34, 581–613 (1992). Zbl 0788.65037, MR 1193013, 10.1137/1034116 |
| Reference:
|
[LIT28] H. Yserentant: On the multi-level splitting of finite element spaces.Numer. Math. 49, 379–412 (1986). Zbl 0625.65109, MR 0853662, 10.1007/BF01389538 |
| Reference:
|
[LIT29] H. Yserentant: Old and new convergence proofs for multigrid methods.Acta Numerica 1, 285–326 (1993). Zbl 0788.65108, MR 1224685, 10.1017/S0962492900002385 |
| Reference:
|
[LIT30] X. Zhang: Multilevel Schwarz methods.Numer. Math. 63, 521–539 (1992). Zbl 0796.65129, MR 1189535, 10.1007/BF01385873 |
| . |
