Conditioning analysis of separate displacement preconditioners for some nonlinear elasticity systems
Mathematics and Computers in Simulation
Preconditioning for a Class of Spectral Differentiation Matrices
Journal of Scientific Computing
Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems
Journal of Computational and Applied Mathematics
A mesh independent superlinear algorithm for some nonlinear nonsymmetric elliptic systems
Computers & Mathematics with Applications
On Superlinear PCG Methods for FDM Discretizations of Convection-Diffusion Equations
Numerical Analysis and Its Applications
Mesh Independent Convergence Rates Via Differential Operator Pairs
Large-Scale Scientific Computing
Preconditioning operators and Sobolevgradients for nonlinear elliptic problems
Computers & Mathematics with Applications
Sobolev gradient preconditioning for the electrostatic potential equation
Computers & Mathematics with Applications
A parallel algorithm for systems of convection-diffusion equations
NMA'06 Proceedings of the 6th international conference on Numerical methods and applications
Journal of Computational and Applied Mathematics
An Analysis of Equivalent Operator Preconditioning for Equation-Free Newton-Krylov Methods
SIAM Journal on Numerical Analysis
From Functional Analysis to Iterative Methods
SIAM Review
On symmetric part PCG for mixed elliptic problems
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
Nonlinear least squares and Sobolev gradients
Applied Numerical Mathematics
Hi-index | 0.00 |
This work is motivated by the preconditioned iterative solution of linear systems that arise from the discretization of uniformly elliptic partial differential equations. Iterative methods with bounds independent of the discretization are possible only if the preconditioning strategy is based upon equivalent operators. The operators A, B: W - V are said to be V norm equivalent if @?Au@?"v@?Bu@?"v is bounded above and below by positive constants for u@e D, where D is ''sufficiently dense.'' If A is V norm equivalent to B, then for certain discretization strategies one can use B to construct a preconditioned iterative scheme for the approximate solution of the problem Au = F. The iteration will require an amount of work that is at most a constant times the work required to approximately solve the problem Bu@^ = \@^tf to reduce the V norm of the error by a fixed factor. This paper develops the theory of equivalent operators on Hubert spaces. Then, the theory is applied to uniformly elliptic operators. Both the strong and weak forms are considered. Finally, finite element and finite difference discretizations are examined.