## On a BPX-preconditioner for P1 elements.(English)Zbl 0787.65018

An optimal multilevel preconditioner for nonconforming P1 elements discretizations of second order elliptic boundary value problems is derived. The resulting condition numbers are uniformly bounded with respect to the number of levels $$j$$ which is known for the conforming case, and improve the previous results for nonconforming P1 elements.

### MSC:

 65F10 Iterative numerical methods for linear systems 65F35 Numerical computation of matrix norms, conditioning, scaling 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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### References:

 [1] Braess, D., Verfürth, R.: Multigrid methods for nonconforming finite element methods. SIAM J. Numer. Anal.27, 979–986 (1990). · Zbl 0703.65067 [2] Bramble, J. H.: Private communication. Summer Conference on Domain Decomposition, Lambrecht 1991. [3] Bramble, J. H., Pasciak, J. E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comp.56, 1–34 (1991). · Zbl 0718.65081 [4] Bramble, J. H., Pasciak, J. E., Xu, J.: Parallel multilevel preconditioners. Math. Comp.55, 1–22 (1990). · Zbl 0703.65076 [5] Brenner, S. C.: An optimal-order multigrid method for P1 nonconforming finite elements. Math. Comp.53, 1–15 (1989). · Zbl 0664.65103 [6] Dörfler, W.: Hierarchical bases for elliptic problems. Math. Comp.58, 513–528 (1992). · Zbl 0766.65088 [7] Dörfler, W.: A note on the preconditioner of Bramble, Pasciak, and Xu. Preprint, Inst. Angew. Math., Univ. Zürich (1991). [8] Dryja, M., Widlund, O. B.: An additive variant of the Schwarz alternating method for the case of many subregions. Tech. Rep.339, Dep. Comp. Science, Courant Institute, New York (1987). [9] Dryja, M., Widlund, O. B.: Multilevel additive methods for elliptic finite element problems. In: Hackbusch, W. (ed.) Parallel Algorithms for PDE. Proc. 6th GAMM-Seminar, Kiel 1990. Braunschweig: Vieweg 1991. · Zbl 0719.65084 [10] Oswald, P.: On a hierarchical basis multilevel method with nonconforming P1 elements, Numer. Math.62, 189–212 (1992). · Zbl 0767.65078 [11] Oswald, P.: On discrete norm estimates related to multilevel preconditioners in the finite element method, In: Constructive theory of functions, Proc. Int. Conf. Varna 1991 (K. G. Ivanov, P. Petrushev, B. Sendov, ed.) pp. 203–214. Bulg. Acad. Sci. Sofia 1992. [12] Oswald, P.: On norm equivalencies multilevel preconditioners adapted to a variable coefficient problem. Fo.-Erg. FSU Jena, Math. 91/1 (1991). [13] Oswald, P.: On function spaces related to finite element approximation theory, Z. Anal. Anwendungen9, 43–64 (1990). · Zbl 0703.41018 [14] Widlund, O. B.: Some Schwarz methods for symmetric and nonsymmetric elliptic problems. Tech. Rep. 581, Dep. Comp. Science, Courant Institute, New York (1991). · Zbl 0772.65025 [15] Xu, J.: Iterative methods by space decomposition and subspace correction, SIAM Review34, 581–613 (1992). · Zbl 0788.65037 [16] Yserentant, H.: Two preconditioners based on the multi-level splitting of finite element spaces. Numer. Math.58, 163–184 (1990). · Zbl 0708.65103 [17] Zhang, X.: Multilevel Schwarz methods. Numer. Math.63, 521–539 (1992). · Zbl 0796.65129
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