Domain Decomposition Methods Based On The Natural Boundary Reduction And Adaptive Methods For Exterior Problems Of Elliptic Equations

The exterior problems of PDEs are widely used in scientific and engineering computing,such as in electromagnetics,acoustics,fluid dynamics,aerodynamics,etc.,so the numerical methods of exterior problems have always been an important research direction of computational mathematics.Under the requirement of ensuring the accuracy of numerical solutions,overcoming the difficulties caused by the boundlessness of computational domains is often the key to the numerical methods of exterior problems.In recent decades,many effective methods have been developed for numerical solution to exterior problems,such as boundary element method,coupling method of boundary element and finite element,artificial boundary method,infinite element method,spectral method and domain decomposition algorithm on unbounded domains,etc.In these methods,boundary reduction is an important idea to deal with the problems of unbounded domains.Based on the principle of natural boundary reduction,this dissertation proposes a domain decomposition method using the curved-FEM and adaptive methods,and uses it to solve anisotropic elliptic exterior problems.The research is divided into the following two parts:In the first part,a Schwarz algorithm based on the natural boundary reduction and the curved-FEM for anisotropic exterior problems is studied.From the perspective of the variational method,the geometric convergence for continuous and discrete cases of the algorithm is proved by using the projection theory,and the error estimates between the iterative solution and the exact solution are given in H~2 and L~2 norms based on the conforming curved-FEM.The error depends on the curved-FEM mesh size and the iteration times.Numerical examples show that,compared with the standard-FEM,Schwarz algorithm using the curved-FEM can keep convergence properties of the iterative solution unchanged and reduce the error between the iterative solution and the exact solution.In the second part,an adaptive D-N alternating method based on the natural boundary reduction and the curved-FEM is studied.Firstly,considering the effect of the number of truncated terms in the natural boundary integral equation of the algorithm,a D-N alternating method with truncated terms is established.Then,the Steklov-Poincaré operator is used to obtain the convergence of the D-N alternating method and its discrete scheme.Based on the conforming curved-FEM,the H~1-estimate between the iterative solution and the exact solution is proved.The error depends on the number of truncated terms,the curved-FEM mesh size,the reduction factor and the iteration times.Numerical examples show that the curved-FEM and the moving mesh method are effective for improving the accuracy of numerical solutions.