A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems

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]  J. Westlake.  Handbook of Numerical Matrix Inversion and Solution of Linear Equations , 1968 .

[2]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[3]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[4]  Walter Mead Patterson,et al.  Iterative methods for the solution of a linear operator equation in Hilbert space - a survey , 1974 .

[5]  R. S. Falk.  An analysis of the finite element method using Lagrange multipliers for the stationary Stokes equations , 1976 .

[6]  P. Swarztrauber.  THE METHODS OF CYCLIC REDUCTION, FOURIER ANALYSIS AND THE FACR ALGORITHM FOR THE DISCRETE SOLUTION OF POISSON'S EQUATION ON A RECTANGLE* , 1977 .

[7]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[8]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[9]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[10]  R. S. Falk,et al.  Error estimates for mixed methods , 1980 .

[11]  James H. Bramble,et al.  The Lagrange multiplier method for Dirichlet’s problem , 1981 .

[12]  Jean-Claude Nédélec,et al.  Éléments finis mixtes incompressibles pour l'équation de Stokes dans ℝ3 , 1982 .

[13]  Claes Johnson,et al.  Analysis of some mixed finite element methods related to reduced integration , 1982 .

[14]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[15]  Michael Vogelius,et al.  Conforming finite element methods for incompressible and nearly incompressible continua , 1984 .

[16]  A boundary parametric approximation to the linearized scalar potential magnetostatic field problem , 1985 .

[17]  J. Pasciak,et al.  An iterative method for elliptic problems on regions partitioned into substructures , 1986 .

[18]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .