Fast Finite Element Method for the Three-Dimensional Poisson Equation in Infinite Domains (Communications in Computational Physics, 2020)
We aim at a fast finite element method for the Poisson equation in three-dimensionalinfinite domains. Both the exterior and strip-tail problems are considered.By introducing a suitable artificial boundary and imposing the exact boundary conditionof Dirichlet-to-Neumann (DtN) type, we reduce the original infinite domain probleminto a truncated finite domain problem. The point is how to efficiently implementthis exact artificial boundary condition. The traditional modal expansion method ishard to apply for the strip-tail problem with a general cross section. We develop a fastalgorithm based on the Padé approximation for the square root function involved inthe exact artificial boundary condition. The most remarkable advantage of our methodis that it is unnecessary to compute the full eigen system associated with the Laplace-Beltramioperator on the artificial boundary. Besides, compared with the modal expansionmethod, the computational cost of the DtN mapping is significantly reduced.We perform a complete numerical analysis on the fast algorithm. Some numerical examplesare presented to demonstrate the effectiveness of the proposed method.