A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. (English) Zbl 0643.65017

Bramble, James H.; Pasciak, Joseph E.

Math. Comput. 50, No. 181, 1-17 (1988).

The paper treats a preconditioned iterative technique for the solution of saddle point problems with applications to equations of elasticity and Stokes.

Reviewer: M.A.Ibiejugba

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
35J20	Variational methods for second-order elliptic equations
76D07	Stokes and related (Oseen, etc.) flows
74B05	Classical linear elasticity

Keywords:

preconditioning; Lagrange multiplier; mixed methods; saddle point problem; symmetric positive definite system; conjugate gradient iteration; equations of elasticity; numerical experiments; Stokes equation

Software:

symrcm; YSMP

References:

[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. · Zbl 0268.65052
[2]	Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179 – 192. · Zbl 0258.65108 · doi: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 (Unpublished.)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. · Zbl 0477.65077
[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. · Zbl 0619.65113 · doi: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. · Zbl 0595.65111
[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. · Zbl 0615.65112
[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. · Zbl 0623.65118
[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). · Zbl 0338.90047
[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. · Zbl 0383.65058
[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. · Zbl 0492.65012
[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. · Zbl 0351.65028
[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). · Zbl 0467.65062
[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. · Zbl 0516.65010
[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. · Zbl 0413.65081
[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. · Zbl 0482.65058
[18]	J.-C. Nédélec, Éléments finis mixtes incompressibles pour l’équation de Stokes dans \?³, Numer. Math. 39 (1982), no. 1, 97 – 112 (French, with English summary). · Zbl 0488.76038 · doi: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. · Zbl 0284.47002
[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.
[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. · Zbl 0582.76028
[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. · Zbl 0358.65088 · doi: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. · Zbl 0383.35057
[24]	Joan R. Westlake, A handbook of numerical matrix inversion and solution of linear equations, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 0155.19901