## Interior penalty preconditioners for mixed finite element approximations of elliptic problems.(English)Zbl 0857.65117

A second-order elliptic problem $-\nabla\cdot k\nabla p=f\quad\text{in}\quad \Omega,\quad p= 0\quad\text{on}\quad \partial\Omega$ is reformulated as a mixed problem by introducing $${\mathbf u}=-k\nabla p$$: $k^{-1}{\mathbf u}+\nabla p=0,\quad \nabla\cdot{\mathbf u}=f.$ This mixed system is to be solved numerically by aid of finite elements of local support and this rises the need for preconditioners for a matrix $$L=BA^{-1}B^*$$, where $$B$$ is a discrete analogue of $$\nabla$$ and $$A$$ relates to the matrix $$k^{-1}$$. This operator is not local, but by employing an interior penalty method by D. N. Arnold [SIAM J. Numer. Anal. 19, 742-760 (1982; Zbl 0482.65060)] a local operator which is spectrally equivalent to $$L$$, may be designed. This may be used for constructing preconditioners.
The method is employed to an application of overlapping domain decomposition methods.

### MSC:

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

Zbl 0482.65060
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### References:

 [1] Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742 – 760. · Zbl 0482.65060 [2] O. Axelsson, Numerical algorithms for indefinite problems, Elliptic Problem Solvers II , Academic Press, Orlando, 1984, pp. 219–232. CMP 17:03 [3] O. Axelsson and P. S. Vassilevski, A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning, SIAM J. Matrix Anal. Appl. 12 (1991), no. 4, 625 – 644. · Zbl 0748.65028 [4] ——, Construction of variable–step preconditioners for inner–outer iterative methods, Iterative Methods in Linear Algebra , North-Holland, Amsterdam, 1992, pp. 1–14. CMP 92:11 [5] Randolph E. Bank, Bruno D. Welfert, and Harry Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math. 56 (1990), no. 7, 645 – 666. · Zbl 0684.65031 [6] Petter E. Bjørstad and Olof B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal. 23 (1986), no. 6, 1097 – 1120. · Zbl 0615.65113 [7] James H. Bramble and Joseph E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp. 50 (1988), no. 181, 1 – 17. , https://doi.org/10.1090/S0025-5718-1988-0917816-8 James H. Bramble and Joseph E. Pasciak, Corrigenda: ”A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems”, Math. Comp. 51 (1988), no. 183, 387 – 388. · Zbl 0643.65017 [8] ——, Iterative techniques for time dependent Stokes problem. Preprint 1994. [9] J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103 – 134. · Zbl 0615.65112 [10] J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), no. 174, 361 – 369. · Zbl 0595.65111 [11] J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems. Preprint 1994. · Zbl 0873.65031 [12] James H. Bramble, Joseph E. Pasciak, and Jinchao Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), no. 193, 1 – 34. · Zbl 0718.65081 [13] Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 581 – 604 (English, with French summary). · Zbl 0689.65065 [14] Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217 – 235. · Zbl 0599.65072 [15] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. · Zbl 0788.73002 [16] L. C. Cowsar, Dual variable Schwarz methods for mixed finite elements, Report TR93-09, Rice University, Houston, 1993. [17] Lawrence C. Cowsar, Jan Mandel, and Mary F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp. 64 (1995), no. 211, 989 – 1015. · Zbl 0828.65135 [18] J. Douglas, Jr. and J. Wang, A new family of mixed finite element spaces over rectangles, Comp. Appl. Math. 12 (1993), pp. 183–197. CMP 94:16 [19] M. Dryja and O. B. Widlund, An additive variant of the Schwarz alternating method for the case of many subregions, Technical Report 339, Courant Institute of Mathematical Sciences, 1987. [20] M. Dryja, B. Smith, and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, Technical Report 638, Courant Institute of Mathematical Sciences, 1993. · Zbl 0818.65114 [21] Howard C. Elman and Gene H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994), no. 6, 1645 – 1661. · Zbl 0815.65041 [22] R. E. Ewing and M. F. Wheeler, Computational aspects of mixed finite element methods, Numerical Methods for Scientific Computing , North-Holland Publishing Co., Amsterdam, 1983, pp. 163–172. [23] R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249 – 277 (English, with French summary). · Zbl 0467.65062 [24] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077 [25] Roland Glowinski and Mary Fanett Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 144 – 172. · Zbl 0661.65105 [26] P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1 – 42. · Zbl 0658.65090 [27] A.M. Matsokin and S. Nepomnyaschikh, On the Schwarz alternating method, Preprint, Computing Center, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 1984. · Zbl 0825.65090 [28] C. C. Paige and M. A. Saunders, Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 617 – 629. · Zbl 0319.65025 [29] Werner Queck, The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type, SIAM J. Numer. Anal. 26 (1989), no. 4, 1016 – 1030. · Zbl 0674.65008 [30] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292 – 315. Lecture Notes in Math., Vol. 606. [31] Torgeir Rusten and Ragnar Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), no. 3, 887 – 904. Iterative methods in numerical linear algebra (Copper Mountain, CO, 1990). · Zbl 0760.65033 [32] Torgeir Rusten and Ragnar Winther, Substructure preconditioners for elliptic saddle point problems, Math. Comp. 60 (1993), no. 201, 23 – 48. · Zbl 0795.65072 [33] David Silvester and Andrew Wathen, Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners, SIAM J. Numer. Anal. 31 (1994), no. 5, 1352 – 1367. · Zbl 0810.76044 [34] P. S. Vassilevski and R. D. Lazarov, Preconditioning saddle-point problems arising from mixed finite element discretization of elliptic problems, Report CAM 92–46, Department of Mathematics, UCLA, 1992. [35] P. S. Vassilevski and J. Wang, An application of the abstract multilevel theory to nonconforming finite element methods, SIAM J. Numer. Anal. 32 (1995), 235–248. CMP 95:07 [36] ——, Multilevel methods for cell–centered finite difference approximations of elliptic problems, Preprint, 1992. [37] R. Verfürth, A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem, IMA J. Numer. Anal. 4 (1984), no. 4, 441 – 455. · Zbl 0563.76028 [38] Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581 – 613. · Zbl 0788.65037
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