Fast Fourier-Galerkin Methods For A Kind Of Transmission Problem

The Galerkin method using the trigonometric polynomial ba-sis(which we call the Fourier basis)among other numerical methods is a standard numerical treatment for solving singular boundary in-tegral equations.In this paper,we call such a method the Fourier-Galerkin method.This method enjoys nice convergence properties but at the same time sufers from having large computational costs.When the order of the linear system n is large,the computational cost needed to set up the coefficient matrix and then to solve the system is large.Therefore,approximating the dense coefficient ma-trix by a sparse matrix is practically important.In this thesis,we investigate a transmission problem for the Laplace equation in a periodic half-plane for modelling the elec-tric potential on an interface between a thin layer of oil and the surrounding air.Our strategy will be to reformulate our trans-mission problem in a periodic half-plane as two boundary integral problems defined only on the interface.First compress this full coefficient matrix into a sparse matrix.Then apply the numerical integration scheme to obtain the discrete truncated linear system with a nearly linear computational cost.At last,the discrete linear system is solved.The optimal convergence order with quasi-linear complexity order of the proposed method are esta,blished through specific numerical examples.This article is organized as follows:Chapter 1 and Chapter 2 present the current research status at home and abroad for the Laplace equation in a periodic half-plane for modelling the electric potential on an interface between a thin layer of oil and the surrounding air.In these two chapters,we will also introduce Green’s functions,single and double layer potential,and Sobolev function space.Chapter 3 introduce the model,using Green’s formulas combined with potential theory to reformulate our transmission problem in a periodic half-plane as two boundary integral problems,after the integral equation is parameterized,a truncating strategy is adopted for the coefficient matrix to obtain a truncated linear equation system.Finally,we consider solving the linear systems by the multilevel augmentation method.Chap-ter 4 will give two numerical examples to confirm the theoretical estimates and to demonstrate the approximation accuracy end ef-ficiency of the proposed algorithm.