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Keywords:
mixed finite elements; multi-level solver
Summary:
We outline a solution method for mixed finite element discretizations based on dissecting the problem into three separate steps. The first handles the inhomogeneous constraint, the second solves the flux variable from the homogeneous problem, whereas the third step, adjoint to the first, finally gives the Lagrangian multiplier. We concentrate on aspects involved in the first and third step mainly, and advertise a multi-level method that allows for a stable computation of the intermediate and final quantities in optimal computational complexity.
References:
[1] I. Babuška: The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973), 179–192. DOI 10.1007/BF01436561 | MR 0359352
[2] S. Brenner: A multigrid algorithm for the lowest order Raviart-Thomas mixed triangular finite element method. SIAM J. Numer. Anal. 29 (1992), 647–678. DOI 10.1137/0729042 | MR 1163350 | Zbl 0759.65080
[3] J. H. Brandts: The Cauchy-Riemann equations: discretization by finite elements, fast solution of the second variable, and a posteriori error estimation. Advances Comput. Math. 15 (2001), 61–77. DOI 10.1023/A:1014217225870 | MR 1887729 | Zbl 0996.65120
[4] Z. Cai, C. I. Goldstein, J. Pasciak: Multilevel iteration for mixed finite element systems with penalty. SIAM J. Sci. Comput. 14 (1993), 1072–1088. DOI 10.1137/0914065 | MR 1232176
[5] F. Brezzi: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8 (1974), 129–151. MR 0365287 | Zbl 0338.90047
[6] R. Ewing, J. Wang: Analysis of multilevel decomposition iterative methods for mixed finite element methods. RAIRO Modèl. Math. Anal. Numér. 28 (1994), 377–398. DOI 10.1051/m2an/1994280403771 | MR 1288504
[7] V. Girault, P. A. Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, Springer, Berlin, 1986. MR 0851383
[8] R. Hiptmair, R. H. W. Hoppe: Multigrid methods for mixed finite elements in three dimensions. Numer. Math. 82 (1999), 253–279. DOI 10.1007/s002110050419 | MR 1685461
[9] R. Hiptmair, T. Schiekofer, B. Wohlmuth: Multilevel preconditioned augmented Lagrangian techniques for 2nd order mixed problems. Computing 57 (1996), 25–48. DOI 10.1007/BF02238356 | MR 1398269
[10] P. A. Raviart, J. M. Thomas: A mixed finite element method for 2nd order elliptic problems. Math. Aspects Finite Elem. Math., Proc. Conf. Rome 1975. Lect. Notes Math. 606 (1977), 292–315. MR 0483555
[11] R. P. Stevenson: A stable, direct solver for the gradient equation. Math. Comp (to appear). MR 1933813 | Zbl 1012.65126
[12] A. J. Wathen, B. Fischer, D. J. Silvester: The convergence rate of the minimal residual method for the Stokes problem. Numer. Math. 71 (1995), 121–134. DOI 10.1007/s002110050138 | MR 1339734
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