Fast Algorithms For The Partial Differential Equations In Infinite Domains

Numerous problems of science and engineering can be modeled by partial differential equations in infinite domains.The artificial boundary method is an effective and popular method for solving this kind of problems.By introducing a suitable artificial boundary and imposing exact or approximate boundary condition,we reduce the original infinite domain problem into a truncated finite domain problem.Some normal numerical methods can then be applied to solve the truncated finite domain problem.In this paper,we first aim at a fast finite element method for the three-dimensional Poisson equation in infinite domains,including the exterior and strip-tail problems.Exact Dirichlet-to-Neumann type artificial boundary conditions are derived to reduce the original infinite domain problems into some truncated domain problems.Based on the Pade approximation and the best relative Chebyshev approximation for the square root function,we develop two fast algorithms to approximate the exact artificial boundary conditions.One remarkable advantage is that it is unnecessary to compute the full eigen system associated with the surface Laplacian operator on the artificial boundaries.In addition,compared with the modal expansion method,the computational cost of the Dirichlet-to-Neumann mapping is significantly reduced.We perform a complete numerical analysis on the fast algorithm and present examples to demonstrate the efficiency of the proposed method.In this paper,we also consider the stability and error analysis of a second-order fast approximation for the one-dimensional local and nonlocal diffusion equations in infinite domains.By using the central difference scheme to discretize the spatial local derivative operator and using asymptotically compatible difference scheme to discretize the spatial nonlocal diffusion operator,and applying second-order backward differentiation formula to the temporal derivative,we derive a fully discrete system involving an infinite number of degrees of freedom.We then obtain a Dirichlet-to-Neumann type artificial boundary conditions from the fully discrete system in a unified framework to reduce the infinity discrete system into a finite discrete system in a truncated computational domain of interest.To arrive at this,we first apply the z-transform and solve an exterior problem by an iteration technique to obtain an exact Dirichlet-to-Dirichlet type artificial boundary conditions.After that,we reformulate the Dirichlet-to-Dirichlet type mapping equivalently as the Dirichlet-to-Neumann type mapping by the Green formula.Based on the Dirichlet-to-Neumann type mapping and some open but reasonable assumptions,we present the stability and convergence analysis of the reduced problem.Based on some approximation of the contour integral induced by the inverse z-transform,we further develop a fast convolution algorithm to efficiently implement the exact artificial boundary conditions.The stability and error analysis are also established,and we provide numerical examples to demonstrate the effectiveness of the proposed approach.