A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems
Authors:
James H. Bramble and Joseph E. Pasciak
Journal:
Math. Comp. 50 (1988), 117
MSC:
Primary 65N30; Secondary 65F10
Corrigendum:
Math. Comp. 51 (1988), 387388.
MathSciNet review:
917816
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Abstract: This paper provides a preconditioned iterative technique for the solution of saddle point problems. These problems typically arise in the numerical approximation of partial differential equations by Lagrange multiplier techniques and/or mixed methods. The saddle point problem is reformulated as a symmetric positive definite system, which is then solved by conjugate gradient iteration. Applications to the equations of elasticity and Stokes are discussed and the results of numerical experiments are given.
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 [1]
 A. K. Aziz & I. Babuška, "Survey lectures on the mathematical foundations of the finite element method, Part I," in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, ed.), Academic Press, New York, 1972, pp. 1362. MR 0421106 (54:9111)
 [2]
 I. Babuška, "The finite element method with Lagrangian multipliers," Numer. Math., v. 20, 1973, pp. 179192. MR 0359352 (50:11806)
 [3]
 J. H. Bramble, Iterative Methods for Solving Finite Element or Finite Difference Equations for Elliptic Problems, Lecture Notes. (Unpublished.)
 [4]
 J. H. Bramble, "The Lagrange multiplier method for Dirichlet's problem," Math. Comp., v. 37, 1981, pp. 112. MR 616356 (83h:65119)
 [5]
 J. H. Bramble & J. E. Pasciak, "A boundary parametric approximation to the linearized scalar potential magnetostatic field problem," Appl. Numer. Math., v. 1, 1985, pp. 493514. MR 814774 (87b:78014)
 [6]
 J. H. Bramble, J. E. Pasciak & A. H. Schatz, "An iterative method for elliptic problems on regions partitioned into substructures," Math. Comp., v. 46, 1986, pp. 361369. MR 829613 (88a:65123)
 [7]
 J. H. Bramble, J. E. Pasciak & A. H. Schatz, "The construction of preconditioners for elliptic problems by substructuring. I," Math. Comp., v. 47, 1986, pp. 103134. MR 842125 (87m:65174)
 [8]
 J. H. Bramble, J. E. Pasciak & A. H. Schatz, "The construction of preconditioners for elliptic problems by substructuring. II," Math. Comp., v. 49, 1987, pp. 116. MR 890250 (88j:65248)
 [9]
 F. Brezzi, "On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrange multipliers," RAIRO, 1974, pp. 129151. MR 0365287 (51:1540)
 [10]
 R. Chandra, Conjugate Gradient Methods for Partial Differential Equations, Yale Univ., Dept. of Comp. Sci., Rep. No. 129, 1978.
 [11]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, New York, 1978. MR 0520174 (58:25001)
 [12]
 S. C. Eisenstat, M. C. Gursky, M. H. Schultz & A. H. Sherman, "Yale sparse matrix package, I. The symmetric codes," Internat. J. Numer. Methods Engrg., v. 18, 1982, pp. 11451151.
 [13]
 R. S. Falk, "An analysis of the finite element method using Lagrange multipliers for the stationary Stokes equations," Math. Comp., v. 30, 1976, pp. 241269. MR 0403260 (53:7072)
 [14]
 R. S. Falk & J. E. Osborn, "Error estimates for mixed methods," RAIRO Numer. Anal., v. 14, 1980, pp. 249277. MR 592753 (82j:65076)
 [15]
 A. George & J. W. Liu, Computer Solution of Large Sparse Positive Definite Systems, PrenticeHall, Englewood Cliffs, N. J., 1981. MR 646786 (84c:65005)
 [16]
 V. Girault & P. A. Raviart, Finite Element Approximation of the NavierStokes Equations, Lecture Notes in Math., vol. 749, SpringerVerlag, New York, 1981. MR 548867 (83b:65122)
 [17]
 C. Johnson & J. Pitkäranta, "Analysis of some mixed finite element methods related to reduced integration," Math. Comp., v. 38, 1982, pp. 375400. MR 645657 (83d:65287)
 [18]
 J. C. Nedelec, "Elements finis mixtes incompressibles pour l'equation de Stokes dans ," Numer. Math., v. 39, 1982, pp. 97112. MR 664539 (83g:65111)
 [19]
 W. M. Patterson, 3Rd, Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space  A Survey, Lecture Notes in Math., vol. 394, SpringerVerlag, New York, 1974. MR 0438701 (55:11609)
 [20]
 P. A. Raviart & J. M. Thomas, "A mixed finite element method for 2nd order elliptic problems," in Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, eds.), Lecture Notes in Math., vol. 606, SpringerVerlag, New York, 1977, pp. 292315. MR 0483555 (58:3547)
 [21]
 L. R. Scott & M. Vogelius, Conforming Finite Element Methods for Incompressible and Nearly Incompressible Continua, Inst, for Phys. Sci. and Tech., Univ. of Maryland, Tech. Rep. BN1018, 1984. MR 818790 (87h:65202)
 [22]
 P. N. Swarztrauber, "The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle," SIAM Rev., v. 19, 1977, pp. 490501. MR 0438732 (55:11639)
 [23]
 R. Temam, NavierStokes Equations, NorthHolland, New York, 1977. MR 0609732 (58:29439)
 [24]
 J. Westlake, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, Wiley, New York, 1968. MR 0221742 (36:4794)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809178168
PII:
S 00255718(1988)09178168
Article copyright:
© Copyright 1988
American Mathematical Society
