A D-N Alternating Algorithm for Exterior Laplace Problem with Mixed Boundary Value Conditions

摘要

This paper develops a Dirichlet-Neumann (D-N) alternating algorithm to solve the mixed boundary value problem of Laplace equation in an infinite domain, and analyses the convergence of the algorithm. By choosing a circle surrounding the original boundary as an artificial boundary to divide the unbounded domain into two sub-domains, we can make use of the natural boundary reduction (NBR) method in the infinite sub-domain to solve a Dirichlet boundary value problem while use the finite element method in the finite sub-domain to solve a mixed boundary value problem. We prove that the algorithm is convergent geometrically for any relaxation factor between 0 and 1. The numerical experiment results also display that the sequence of iterative solutions is geometrically convergent, the convergence rate is independent of the finite element mesh size h, and the maximum nodal error on Ω_1 is roughly of O(h~2).