The D-N alternating algorithm based on the curved-FEM and moving mesh for the exterior harmonic problem

Abstract

Abstract In this paper, we study the numerical solution of the exterior harmonic problem with the Dirichlet boundary value condition. Owing to the difficulties of solving directly in the unbounded domain, we decompose the exterior problem into a mixed boundary value problem on a bounded annular subdomain and a Dirichlet boundary value problem on an unbounded subdomain by constructing an artificial boundary. Then, the Dirichlet-Neumann (D-N) alternating algorithm is proposed to solve two sub-problems alternately, where the former is solved by the curved-FEM, while the latter is solved by the principle of natural boundary reduction (NBR). Rather than the standard FEM, the curved-FEM is conforming and gives better discrete approximate variational formulation of the algorithm. The geometrical convergence of the discrete algorithm is also obtained. Finally, the D-N alternating algorithm based on the curved-FEM and the moving mesh method is shown to achieve higher precision in the numerical example.