Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations

We consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself-adjoint elliptic boundary value problems. We derive bounds on convergence rates that are independent of discretization mesh size. For two-dimensional problems on rectangular domains, discretized on an $n \times n$ grid, these bounds lead to asymptotic operation counts of $O(n^2 \log n\log \varepsilon ^{ - 1} )$ to achieve relative error $\varepsilon $ and $O(n^2 (\log n)^2 )$ to reach truncation error.

[1]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[2]  M. Lees.  A priori estimates for the solutions of difference approximations to parabolic partial differential equations , 1960 .

[3]  George E. Forsythe,et al.  Finite-Difference Methods for Partial Differential Equations , 1961 .

[4]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[5]  J. Daniel.  On the approximate minimization of functionals , 1969 .

[6]  F. Dorr.  The Direct Solution of the Discrete Poisson Equation on a Rectangle , 1970 .

[7]  H. Kitamura.  BookThe emerging Japanese superstate: Challenge and response: by Herman Kahn 274 pages. $7.95. Prentice-Hall, Inc, Englewood Cliffs, NJ , 1971 .

[8]  Olof B. Widlund,et al.  On the use of Fast Methods for Separable Finite Difference Equations for the Solution of General Elliptic Problems , 1972 .

[9]  G. Golub,et al.  Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. , 1972 .

[10]  P. Swarztrauber.  A direct Method for the Discrete Solution of Separable Elliptic Equations , 1974 .

[11]  R. Bank.  Marching Algorithms for Elliptic Boundary Value Problems. II: The Variable Coefficient Case , 1977 .

[12]  D. Rose,et al.  Marching Algorithms for Elliptic Boundary Value Problems. I: The Constant Coefficient Case , 1977 .

[13]  O. Widlund.  A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations , 1978 .

[14]  Roland A. Sweet,et al.  Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3] , 1979, TOMS.

[15]  O. Axelsson.  Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations , 1980 .

[16]  Kang C. Jea,et al.  Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods , 1980 .

[17]  G. Hedstrom,et al.  The effect of cell reynolds number on the computation of a boundary layer , 1980 .

[18]  Y. Saad.  Krylov subspace methods for solving large unsymmetric linear systems , 1981 .

[19]  J. Strikwerda.  A Generalized Conjugate Gradient Method for Non-Symmetric Systems of Linear Equations. , 1981 .

[20]  On the Eigenvalues of Second Order Elliptic Difference Operators , 1982 .

[21]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .