A new scheme to obtain pollution-free solution for plane waves and to generate a continuous Petrov–Galerkin method for the Helmholtz problem

Abstract

In this paper, we propose a new scheme to solve the Helmholtz problem. The proposed scheme has the ability to obtain a pollution-free solution and with good accuracy, when the exact solution is a plane wave or a superposition of plane waves with the same direction of propagation. This ability is valid for all the directions of plane waves. The scheme can be naturally extended to include other types of solutions of the homogeneous Helmholtz equation, as for example: cylindrical waves. The presented results confirm the ability described previously to obtain a pollution-free solution for all the directions of propagation of the plane waves. Also in this paper we propose a new continuous Petrov–Galerkin method with the goal to analyze Helmholtz problems where the solution is not a plane wave or a superposition of plane waves. This method was applied to non-homogeneous Helmholtz problems having good accuracy.