Abstract
The nonoverlapping domain decomposition method, which is based on the natural boundary reduction, is applied to solve the exterior Helmholtz problem over a three-dimensional domain. The basic idea is to introduce a spherical artificial boundary; the original unbounded domain is changed into a bounded subdomain and a typical unbounded region; then, a Dirichlet-Nuemann (D-N) alternating method is presented; the finite element method and natural boundary element methods are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. The convergence of the D-N alternating algorithm and its discretization are studied. Some numerical experiments are presented to show the performance of this method.
Highlights
Many scientific and engineering problems can be reduced to exterior boundary value problems of partial differential equations
The artificial boundary Γ1 divides the original unbounded region into two subregions Ωext and Ωint, where Ωext is the outside subregion and Ωint is the bounded subregion surrounded by ΓR and Γ1, Ωext and Ωint are nonoverlaping domains
Let n = n + 1, and go to Step 2, where u1n and u2n are the nth approximate solutions in Ωint and Ωext, respectively. n1 and n2 denote the unit outward normals of Γ1 with respect to two neighboring subdomains; θn denotes the nth relaxation factor and λ0 is an arbitrary function in H1/2(Γ1)
Summary
Many scientific and engineering problems can be reduced to exterior boundary value problems of partial differential equations. The numerical methods for solving boundary value problems, such as the finite element method and the finite difference method, are very effective on a bounded domain, yet we often find it difficult to be applied to unbounded problem directly. To solve such problems in infinite region numerically, there are a variety of numerical methods, cf [1–3] and references therein for more details. Jia et al [7] investigated the coupled natural boundary element-finite element method for solving 3D exterior Helmholtz problem. Some numerical examples are presented to illustrate the feasibility and efficiency of this method
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