A Weighted Variational Formulation Based on Plane Wave Basis for Discretization of Helmholtz Equations

In this paper we are concerned with numerical methods for solving Helmholtz equations. We propose a new variant of the Variational Theory of Complex Rays (VTCR) method introduced in [15, 16]. The approximate solution generated by the new variant has higher accuracy than that generated by the original VTCR method. Moreover, the accuracy of the resulting approximate solution can be further increased by adding two suitable positive relaxation parameters into the new variational formula. Besides, a simple domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Numerical results confirm the efficiency of the new method.