Equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions. (English) Zbl 1198.65225
Summary: The numerical solution of linear elliptic partial differential equations often involves finite element discretization, where the discretized system is usually solved by some conjugate gradient method. The crucial point in the solution of the obtained discretized system is a reliable preconditioning, that is to keep the condition number of the systems under control, no matter how the mesh parameter is chosen. The preconditioned conjugate gradient method is applied to solving convection-diffusion equations with nonhomogeneous mixed boundary conditions. Using the approach of equivalent and compact-equivalent operators in Hilbert space, it is shown that for a wide class of elliptic problems the superlinear convergence of the obtained preconditioned conjugate gradient method is mesh independent under finite element method discretization.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65F08 | Preconditioners for iterative methods |
| 65F10 | Iterative numerical methods for linear systems |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
conjugate gradient method; preconditioning; equivalent operators; operator pairs; superlinear convergence; mesh independence; numerical examples; linear elliptic equations; finite element; convection-diffusion equations; nonhomogeneous mixed boundary conditionsReferences:
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