| There are many numerical methods to solve PDE in unbounded domains of scientific research and engineering computation. NBEM and the coupling method of NBEM-FEM can keep the original boundary value problem for many useful properties of the advantages, the NBEM with uniform grids on the boundary and the coupling method of NBEM-FEM have been widely used. For some problems with large gradients or discontinuities, uniform grid is a waste of computing resources, however, the adaptive method can obtain high accurate numerical solution according to the characteristics of PDE. At present, there are three main adaptive methods: p-method, h-method and r-method.The main content of this paper is divided into two parts:In the first part, we propose NBEM and the coupling method of NBEM-FEM for anisotropic external problem with elliptic boundary on unbounded domain, and introduces the moving grid technique on the principle of equal distribution, the corresponding error estimates and convergence theorems are proven, numerical examples verify the convergence theorems and demonstrate the advantage in accuracy and efficiency of the proposed methods; In the second part, we propose an adaptive coupling method for solving anisotropic elliptic PDE in unbounded domains. Firstly, the existence and the uniqueness of the solution for the coupling method are proven, and the a priori error estimates in H’-norm and L2-norm that depend on the size of FEM mesh, the location of the elliptic artificial boundary and truncation of the infinite series in the artificial boundary integral condition are derived. Secondly, the a posteriori error estimates and the a posterior error indicator of the coupling method are obtained. Finally, the adaptive coupling method refines the mesh distribution by the arc-length equidistribution principle and the a posterior error indicator successively. Numerical examples confirm the advantage in accuracy and efficiency for the proposed method. |
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