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), 1-17

MSC:
Primary 65N30; Secondary 65F10

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917816-8

Corrigendum:
Math. Comp. **51** (1988), 387-388.

MathSciNet review:
917816

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[2]**Ivo Babuška,*The finite element method with Lagrangian multipliers*, Numer. Math.**20**(1972/73), 179–192. MR**0359352**, https://doi.org/10.1007/BF01436561**[3]**J. H. Bramble,*Iterative Methods for Solving Finite Element or Finite Difference Equations for Elliptic Problems*, Lecture Notes. (Unpublished.)**[4]**James H. Bramble,*The Lagrange multiplier method for Dirichlet’s problem*, Math. Comp.**37**(1981), no. 155, 1–11. MR**616356**, https://doi.org/10.1090/S0025-5718-1981-0616356-7**[5]**James H. Bramble and Joseph E. Pasciak,*A boundary parametric approximation to the linearized scalar potential magnetostatic field problem*, Appl. Numer. Math.**1**(1985), no. 6, 493–514. MR**814774**, https://doi.org/10.1016/0168-9274(85)90034-0**[6]**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. MR**829613**, https://doi.org/10.1090/S0025-5718-1986-0829613-0**[7]**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. MR**842125**, https://doi.org/10.1090/S0025-5718-1986-0842125-3**[8]**J. H. Bramble, J. E. Pasciak, and A. H. Schatz,*The construction of preconditioners for elliptic problems by substructuring. II*, Math. Comp.**49**(1987), no. 179, 1–16. MR**890250**, https://doi.org/10.1090/S0025-5718-1987-0890250-4**[9]**F. Brezzi,*On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge**8**(1974), no. R-2, 129–151 (English, with loose French summary). MR**0365287****[10]**R. Chandra,*Conjugate Gradient Methods for Partial Differential Equations*, Yale Univ., Dept. of Comp. Sci., Rep. No. 129, 1978.**[11]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[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. 1145-1151.**[13]**Richard S. Falk,*An analysis of the finite element method using Lagrange multipliers for the stationary Stokes equations*, Math. Comput.**30**(1976), no. 134, 241–249. MR**0403260**, https://doi.org/10.1090/S0025-5718-1976-0403260-0**[14]**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). MR**592753****[15]**Alan George and Joseph W. H. Liu,*Computer solution of large sparse positive definite systems*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1981. Prentice-Hall Series in Computational Mathematics. MR**646786****[16]**V. Girault and P.-A. Raviart,*Finite element approximation of the Navier-Stokes equations*, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR**548867****[17]**Claes Johnson and Juhani Pitkäranta,*Analysis of some mixed finite element methods related to reduced integration*, Math. Comp.**38**(1982), no. 158, 375–400. MR**645657**, https://doi.org/10.1090/S0025-5718-1982-0645657-2**[18]**J.-C. Nédélec,*Éléments finis mixtes incompressibles pour l’équation de Stokes dans 𝑅³*, Numer. Math.**39**(1982), no. 1, 97–112 (French, with English summary). MR**664539**, https://doi.org/10.1007/BF01399314**[19]**Walter Mead Patterson III,*Iterative methods for the solution of a linear operator equation in Hilbert space–a survey*, Lecture Notes in Mathematics, Vol. 394, Springer-Verlag, Berlin-New York, 1974. MR**0438701****[20]**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. MR**0483555****[21]**L. R. Scott and M. Vogelius,*Conforming finite element methods for incompressible and nearly incompressible continua*, Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., 1983) Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 221–244. MR**818790****[22]**Paul 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.**19**(1977), no. 3, 490–501. MR**0438732**, https://doi.org/10.1137/1019071**[23]**Roger Temam,*Navier-Stokes equations. Theory and numerical analysis*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Studies in Mathematics and its Applications, Vol. 2. MR**0609732****[24]**Joan R. Westlake,*A handbook of numerical matrix inversion and solution of linear equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0221742**

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DOI:
https://doi.org/10.1090/S0025-5718-1988-0917816-8

Article copyright:
© Copyright 1988
American Mathematical Society