The Study Of Perfectly Matched Layers Method For Heat Equation In Unbounded Domains

The numerical solution of the heat equation in unbounded domains is studied in this thesis.Specifically,the main contents of this thesis are as follows:In chapter one,the background and the significance of the heat equation in unbounded domains are introduced,and the progress of this field is summarized in both domestic and international.In chapter two,some relevant theories about this thesis are given.They mainly include the definition and the properties of the Laplace transform,the finite element method for heat equation in bounded domains and the Perfectly Matched Layers(PML)method for Helmholtz equation.In chapter three,analogous to the PML method in wave equations,based on the Laplace transform,the PML formulations about the heat equation are obtained by introducing some auxiliary variables in this thesis.Besides,the stability of the PML formulations are analyzed.It is shown that the PML formulations in one space dimension and three space dimensions are strongly stable.The PML formulation in two space dimensions is weakly stable.Meanwhile,one property of the auxiliary function is obtained.The value of the auxiliary function in computation region is zero forever,while the value is not zero in the absorbing layers.The validity of the PML method for the heat equation is verified by numerical experiments.At the end of this chapter,the influence factors of the error are analyzed using numerical experiments.For the absorbing function of power function type,the numerical results show that the influence are as follows.(1)The order of the absorbing function has little effect on the error.(2)There is an optimal absorbing strength of the absorbing function.When the absorbing strength is smaller than the optimal value,the error decrease with the increase of the absorbing strength.Whereas,when the absorbing strength is bigger than the optimal value,the error increase slightly with the increase of the absorbing strength.(3)If the thickness of the absorbing layer is lager,then the error is smaller.(4)When the thickness of the absorbing layer is a constant,the number of the absorbing layers have little effect on the error.(5)When the PML formulation is discretized by finite element method,the error of the linear basis function is slightly bigger than the quadratic basis function,so the form of the basis function makes little difference to the error.In chapter four,analogous to the PML method in heat equation,the PML method is generalized to the convection diffusion equation,and the PML formulation about the convection diffusion equation is achieved.The validity of the PML method used in the convection diffusion equation also is verified by numerical experiments.In chapter five,the summary of our work and the future work are introduced.