×

zbMATH — the first resource for mathematics

Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods. (English) Zbl 0833.65131
The mixed finite element discretization of second-order elliptic boundary value problems is considered. The efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations is studied. The authors present two multilevel solvers. The first one is based on ideas from the domain decomposition approach. A multilevel preconditioned conjugate gradient iteration acting on an appropriate subspace with a hierarchical type preconditioner that can be derived by subspace decomposition is presented.
The second algorithm is obtained from the mixed hybridization and an alternative adaptive multilevel method based on this technique is derived. The local refinement of the underlying triangulations is done by efficient and reliable a posteriori error estimators which can be obtained by a defect correction in higher order ansatz spaces or by taking into account the superconvergence results.
The performance of both algorithms is illustrated by two test examples. The solution of the first problem has a peak in an internal point while the solution of the second example has boundary layers.
Reviewer: K.Georgiev (Sofia)

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Software:
PLTMG
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] 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). · Zbl 0567.65078 · eudml:193443
[2] I. Babuska, W.C. Rheinboldt: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736-754 (1978). · Zbl 0398.65069
[3] I. Babuska, W.C. Rheinboldt: A posteriori error estimates for the finite element method. Int. J. Numer. Methods Eng. 12, 1597-1615 (1978). · Zbl 0396.65068
[4] R.E. Bank: PLTMG-A Software Package for Solving Elliptic Partial Differential Equations. User’s Guide 6.0. SIAM, Philadelphia, 1990. · Zbl 0717.68001
[5] 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.
[6] R.E. Bank, A. Weiser: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985). · Zbl 0569.65079 · doi:10.2307/2007953
[7] 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. · Zbl 0863.65069
[8] F. Bornemann, H. Yserentant: A basic norm equivalence for the theory of multilevel methods. Numer. Math. 64, 445-476 (1993). · Zbl 0796.65135
[9] D. Braess, R. Verfürth: A posteriori error estimators for the Raviart-Thomas element. Ruhr-Universität Bochum, Fakultät für Mathematik, Bericht Nr. 175, 1994. · Zbl 0866.65071
[10] 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 · doi:10.1137/0729042
[11] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer, Berlin-Heidelberg-New York, 1991. · Zbl 0788.73002
[12] Z. Cai, C.I. Goldstein, J. Pasciak: Multilevel iteration for mixed finite element systems with penalty. SIAM J. Sci. Comput. 14, 1072-1088 (1993). · Zbl 0809.65114 · doi:10.1137/0914065
[13] L.C. Cowsar: Domain decomposition methods for nonconforming finite element spaces of Lagrange-type. Rice University, Houston. Preprint TR 93-11, 1993.
[14] P. Deuflhard, P. Leinen, H. Yserentant: Concepts of an adaptive hierarchical finite element code. IMPACT Comput. Sci. Engrg. 1, 3-35 (1989). · Zbl 0706.65111 · doi:10.1016/0899-8248(89)90018-9
[15] 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. · Zbl 0770.65084
[16] 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). · Zbl 0765.65104 · eudml:193683
[17] 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). · Zbl 0823.65035 · eudml:193744
[18] 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. · Zbl 0359.73007
[19] 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. · Zbl 0843.65075
[20] 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. · Zbl 0889.65124 · doi:10.1137/S0036142994276992
[21] 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.
[22] P. Oswald: On a BPX-preconditioner for P1 elements. Computing 51, 125-133 (1993). · Zbl 0787.65018 · doi:10.1007/BF02243847
[23] 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. · Zbl 0875.65090
[24] P.S. Vassilevski, J. Wang: Multilevel iterative methods for mixed finite elements discretizations of elliptic problems. Numer. Math. 63, 503-520 (1992). · Zbl 0797.65086 · doi:10.1007/BF01385872 · eudml:133692
[25] R. Verfürth: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Manuscript, 1993. · Zbl 0853.65108
[26] 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.
[27] J. Xu: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581-613 (1992). · Zbl 0788.65037 · doi:10.1137/1034116
[28] H. Yserentant: On the multi-level splitting of finite element spaces. Numer. Math. 49, 379-412 (1986). · Zbl 0625.65109 · doi:10.1007/BF01389672 · eudml:133143
[29] H. Yserentant: Old and new convergence proofs for multigrid methods. Acta Numerica 1, 285-326 (1993). · Zbl 0788.65108
[30] X. Zhang: Multilevel Schwarz methods. Numer. Math. 63, 521-539 (1992). · Zbl 0796.65129 · doi:10.1007/BF01385873 · eudml:133693
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.