Summary
We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.
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Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J. (1991): Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comput.57, 1–21
Brezzi, F., Fortin, M. (1991): Mixed and hybrid finite element methods. Springer, Berlin Heidelberg New York
Cai, X.C., Widlund, O.B. (1992): Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Stat. Comput.13
Concus, P., Golub, G.H., Meurant, G. (1985): Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comput.6, 220–252
Cowsar, L.E., Wheeler, M.F. (1991): Parallel domain decomposition method for mixed finite elements for elliptic partial differential equations. In: R. Glowinski, Y.A. Kuznetsov, G. Meurant, J. Periaux, O.B. Widlund. Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia
Dryja, M., Widlund, O.B. (1989): Some Domain Decomposition Algorithms for Elliptic Problems. In: D.M. Young, Jr. ed., Proceedings of the Conference on Iterative Methods for Large Linear Systems held in Austin, Texas, October 1988, to celebrate the Sixty-fifth Birthday of D.M. Young, Jr., Academic Press, Orlando
Ewing, R.E., Lazarov, R.D., Russel, T.F., Vassilevski, P.S. (1990): Local refinement via domain decomposition techniques for mixed finite element methods with rectangular Raviart-Thomas elements, In: T. Chan, R. Glowinski, J. Périaux, O. Widlund eds., Domain Decomposition Methods, SIAM, Philadelphia
Ewing, R.E., Wang, J. (1991): Analysis of the Schwarz algorithm for mixed finite element methods, Math. Modelling26(6), 739–756
Fortin, M., Aboulaich, R. (1988): Schwarz's Decomposition method for incompressible flow problems, In: R. Glowinski, G.H. Golub, G.A. Meurant, J. Périaux eds., First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia
Girault, V., Raviart, P.A. (1986): Finite element approximation of the Navier-Stokes equations: theory and algorithms. Springer, Berlin Heidelberg New York
Glowinski, R., Kinton, W., Wheeler, M.F. (1990): Acceleration of domain decomposition algorithms for mixed finite elements by multi-level methods. In: T.F. Chan, R. Glowinski, J. Périaux, O.B. Widlund eds., Third International Symposium on Domain Decomposition Methods for Partial Differential Equations. pp. 263–289, SIAM, Philadelphia
Glowinski, R., Le Tallec, P. (1990): Augmented lagrangian methods. SIAM
Glowinski, R., Wheeler, M.F. (1988): Domain decomposition and mixed finite element methods for elliptic problems, In: R. Glowinski, G.H. Golub, G.A. Meurant, J. Périaux eds., First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia
Golub, G.H., Van Loan, C.F. (1989): Matrix computations, Johns Hopkins University Press, Second Edition
Greenbaum, A., Li, C., Han, Z.C. (1989): Parallelizing preconditioned conjugate gradient algorithms. In: Comput. Phys. Commun.53, 295–309
Hart, L., McCormick, S. (1987): Asynchronous multilevel adaptive methods for solving partial differential equations on multiprocessors: computational analysis. Comput. Math. Group. University of Colorado, Denver, (Submitted to Parallel Computing)
Lions, P.L. (1978): Interprétation stochastique de la méthode alternée de schwarz. Academic Press, Paris,268, 325–328
Lions, P.L. (1988): On the Schwarz alternating method. I., In: R. Glowinski, G.H. Golub, G.A. Meurant, J. Périaux eds., First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia
Lions, P.L. (1989): On the Schwarz alternating method. II. In: T. Chan, R. Glowinski, J. Périaux, O. Widlund eds., Domain Decomposition Methods, SIAM, Philadelphia
Mandel, J., McCormick, S. (1989): Iterative solution of elliptic equations with refinement: the two-level case. In: T. Chan, R. Glowinski, J. Périaux, O. Widlund eds., Domain Decomposition Methods, SIAM, Philadelphia
Mathew, T.P. (1992): Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, Part II: Theory65
Mathew, T.P. (1989): Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems. Computer Science Department, Courant Institute of Mathematical Sciences,463 Courant Institute doctoral dissertation
McCormick, S., Thomas, J. (1986): The fast adaptive composite grid (FAC) method for elliptic equations. Math. Comput,46 (1/4), 439–456.
Meijerink, J.A., Van der Vorst, H.A. (1977): An iterative method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput.31(137), 148–162
Miller, K. (1965): Numerical analogs of the Schwarz alternating procedure. Numer. Math.7, 91–103.
Raviart, P.A., Thomas, J.M. (1975): A mixed finite element method for 2-nd order elliptic problems. In: A. Dold, B. Eckmann eds., Mathematical aspects of finite element methods. Lecture Notes of Mathematics,606 Springer, Berlin Heidelberg New York
Roberts, J.E., Thomas, J.-M. (1991): Mixed and hybrid methods. In: P.G. Ciarlet, J.L. Lions eds., Handbook of numerical analysis, Vol. II—finite element methods, Part I, pp. 523–639, North-Holland, Amsterdam
Rusten, T., Winther, R. (1990): A preconditioned iterative method for saddle point problems. SIAM J. Sci. Stat. Comput., Proceedings of Copper Mountain Conference on Iterative Methods
Schwarz, H.A. (1890): Gesammelete Mathematische Abhandlungen,2, 133–143, Springer, Berlin Heidelberg New York, First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, (1870)15, 272–286.
Tang, W.P. (1988): Schwarz splitting and template operators. Ph.D. Thesis, Department of Computer Science, Stanford
Temam, R. (1977): Navier-Stokes equations, North Holland
Wheeler, M.F., Gonzalez, R. (1984): Mixed finite element methods for petroleum reservoir engineering problems, In: R. Glowinski, J.L. Lions. eds., Computing methods in applied sciences and engineering, VI, pp. 639–658, North-Holland, New York
Widlund, O.B. (1989): Optimal iterative refinement methods, Department of Computer Science, Courant Institute. In: T. Chan, R. Glowinski, J. Périaux, O. Widlund eds., Domain decomposition methods, SIAM, Philadelphia
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This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by the Army Research Office under Grant DAAL 03-91-G-0150, while the author was a Visiting Assistant Researcher at UCLA
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Mathew, T.P. Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: Algorithms and numerical results. Numer. Math. 65, 445–468 (1993). https://doi.org/10.1007/BF01385762
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DOI: https://doi.org/10.1007/BF01385762