Generalized finite element method for Helmholtz equation

Abstract

This dissertation presents the Generalized Finite Element Method (GFEM) for the scalar Helmholtz equation, which describes the time harmonic acoustic wave propagation problem. We introduce several handbook functions for the Helmholtz equation, namely the planewave, wave-band, and Vekua functions, and we use these handbook functions to enrich the Finite Element space via the Partition of Unity Method to create the GFEM space. The enrichment of the approximation space by these handbook functions reduces the pollution effect due to wave number and we are able to obtain a highly accurate solution with a much smaller number of degrees-of-freedom compared with the classical Finite Element Method. The q-convergence of the handbook functions is investigated, where q is the order of the handbook function, and it is shown that asymptotically the handbook functions exhibit the same rate of exponential convergence. Hence we can conclude that the selection of the handbook functions from an admissible set should be dictated only by the ease of implementation and computational costs. Another issue addressed in this dissertation is the error coming from the artificial truncation boundary condition, which is necessary to model the Helmholtz problem set in the unbounded domain. We observe that for high q, the most significant component of the error is the one due to the artificial truncation boundary condition. Here we propose a method to assess this error by performing an additional computation on the extended domain using GFEM with high q.