Numerical Algorithms For Quasilinear Equations In Unbounded Domain

In scientific and engineering calculation,the boundary value problem of partial differential equation exists widely in unbounded region.Because of unbounded region,the numerical methods of solving differential equation such as finite element method and difference method usually encounter many difficulties,which are not applicable.Therefore,some new computational methods have emerged,such as the infinite element method and the boundary element method.The boundary element method(BEM)is a numerical method for partial differential equations based on the classical boundary integral equation method and the finite element discrete technology.The natural boundary element method(NBEM)was initiated and developed by Chinese scholars Feng Kang and Yu Dehao.Compared with the classical boundary element method(BEM),this method has its own unique advantages and can solve the problems in unbounded regions accurately and effectively.In recent years,non-linear problems have also attracted much attention.Quasilinear and non-linear problems of non-linear operators with special properties have also been extensively studied.Based on the principle of natural boundary reduction,this thesis studies several numerical methods for solving boundary value problems of quasilinear equations in unbounded domains,which are mainly divided into five chapters:In Chapter 1,the development history of the natural boundary element method and the research status at home and abroad,as well as the related knowledge of Sobolev space,are introduced.In Chapter 2,the principle of natural boundary element method for quasilinear problems is explained in detail,then Poisson integral formula and natural integral equation in typical domains are given,including the upper half plane region,the inner region of the circle(the circle with unit circle and radius R),the outer region(the circle with unit circle and radius R),the elliptic region,the arc or crack region,the elliptic arc region.After that,taking the upper half plane as an example.Three solutions of Poisson integral formula and natural integral equation are given.Finally,the natural integral operator and the natural integral equation are deeply studied.In Chapter 3,the quasilinear problem of anisotropy is studied.The method is the coupling of the natural boundary element and the finite element.According to the principle of boundary reduction,the natural integral equation of artificial boundary is obtained.Then it is transformed into the corresponding variational problem and approximated by finite element method.Then the convergence of the approximate solution is proved.Finally,two numerical examples are given to illustrate the effectiveness of the method.In Chapter 4,the quasilinear equation problem in semi-unbounded domain is studied.The method is non-overlapping domain decomposition algorithm.According to the principle of natural boundary reduction,the natural integral equation on artificial boundary is obtained,and the corresponding alternating algorithm is given.Then the algorithm is discretized and its convergence is analyzed.Then,the proof that the convergence speed of the algorithm is independent of the mesh size of the finite element method and that the D-N algorithm is equivalent to the Richardson iterative algorithm is given.Finally,two numerical examples are selected to demonstrate the and effectiveness of the method.In Chapter 5,the research results and prospects of this thesis are summarized.