Adaptive Method For Solving Elliptic PDEs With The Coupling Of NBEM-FEM

There are many numerical methods to solve PDEs in unbounded domains in scientific research and engineering computation recently. Based on the natural boundary reduction, the natural BEM has the advantage of keeping the original boundary value problem many useful properties, and the method on uniform grid has got a certain application. But for some problems with solution of discontinuity or large gradient, uniform mesh will waste computing resources greatly, so that adaptive mesh method which can achieve high accurate numerical solution on the basis of the character of the PDE is an effective tool. Currently, the main two kinds of adaptive mesh method are as follows:h-method and r-method. The local refining and coarsening mesh method (h-method) improves the accuracy of the numerical solution by increasing the number of grid points and changing the size of grid. Moving mesh method (r-method) changes the position of the grid points to present adaptive method. In the process of solving problems, r-method can keep the topological structure among grid nodes fixed, and move more grid nodes to the place where the solution is required to approximate subtly by using the equidistributed grading function to improve the accuracy of numerical solution. Furthermore, the ellipse and the ellipsoid instead of the circle and sphere are considered as the artificial boundary for the problems outside slender obstacles to save computational cost.In this paper, we take Poisson equation in two-dimensional unbounded domains for example, the main content is divided into two parts:In the first part, we come up with natural BEM and the coupling method of NBE-FE with non-uniform grids on circular boundary, present the corresponding convergence and error estimates respectively, discuss their moving mesh methods based on equidistribution principle, the numerical results prove the theoretical result; In the second part, we discuss a natural BEM with non-uniform grids on elliptic boundary and its coupling method, present the corresponding error estimates, propose examples of moving mesh methods based on equidistribution principle for the above two methods, the numerical results also prove the theoretical result.