Galerkin and Least-Squares Finite Element Processes for 2-D Helmholtz Equation in h, p, k Framework

Abstract

Mathematical and computational details of the finite element processes using the Galerkin method, Galerkin method with weak form, and least-squares processes (LSP) for 2-D Helmholtz equation over a square domain with Dirichlet and Neumann boundary conditions are presented in h, p, k mathematical and computational framework. The order of the approximation space, k, defines the order of the global differentiability (k − 1) of the approximations, and it is an independent parameter, in addition to h and p, in the finite element computations. The discussion of variational consistency (VC) and variational inconsistency (VIC) of the integral forms resulting from the Galerkin method, the Galerkin method with weak form and least-squares processes and their impact on the resulting computational processes are presented. Higher-order global differentiability of the approximations is necessitated by the higher-order global differentiability of the theoretical solutions when they are analytic. Such solutions can be accurately simulated in h, p, k framework up to any desired order global differentiability. Numerical studies are presented for solutions of various classes using the Galerkin method with weak form and least square processes in h, p, k framework. Numerical studies demonstrate various features of the finite element processes and substantiate these observations.

Keywords:

• Galerkin Method/Weak Form
• Least-Squares Processes
• Variational Consistency
• Higher-Order Spaces
• 2-D Helmholtz Equation

Acknowledgments

This work is supported by DEPSCoR, AFOSR and WPAFB under grant numbers F 49620-03-1-0298 and F 49620-03-1-0201.

This research work has been sponsored by grants from DEPSCOR/AFOSR (to Dr. Surana) and ARO (to Dr. Reddy) under grant numbers F-49620-03-1-0298 and 45508-EG, respectively. The support provided by first and third authors' endowed professorship funds is also greatly appreciated.