Superlinear PCG algorithms: Symmetric part preconditioning and boundary conditions. (English) Zbl 1145.65034
Summary: The superlinear convergence of the preconditioned conjugate gradient (PCG) method is studied for nonsymmetric elliptic problems (convection-diffusion equations) with mixed boundary conditions. A mesh independent rate of superlinear convergence is given when symmetric part preconditioning is applied to the finite element discretizations of the boundary value problem. This is the extension of a similar result of the author for Dirichlet problems. The discussion relies on suitably developed Hilbert space theory for linear operators.
MSC:
| 65J10 | Numerical solutions to equations with linear operators |
| 47A50 | Equations and inequalities involving linear operators, with vector unknowns |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
conjugate gradient method; mesh independence; mixed boundary conditions; preconditioning; superlinear convergence; linear operator equation; Hilbert space; nonsymmetric elliptic problems; convection-diffusion equations; symmetric part preconditioningReferences:
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