Equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions

Authors:
Tamás Kurics
Affiliations:
Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117 Budapest, Hungary
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

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 PCG 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 CGM is mesh independent under FEM discretization.