Generalized plane wave discontinuous Galerkin methods for nonhomogeneous Helmholtz equations with variable wave numbers

ABSTRACT

In this paper we are concerned with numerical method for nonhomogeneous Helmholtz equations with variable wave numbers. We derive generalized plane wave basis functions for three-dimensional homogeneous Helmholtz equations with variable wave numbers. Then, by combining the local spectral element method, we design a generalized plane wave discontinuous Galerkin method for the discretization of such nonhomogeneous Helmholtz equations (in both dimensions two and three dimensions). We show that the approximation solutions generated by the proposed discretization method yield error estimates with high accuracies. Numerical results indicate that the resulting approximate solutions generated by the new method possess high accuracy and verify the validity of the theoretical results.

KEYWORDS:

• Helmholtz equation
• nonhomogeneous
• variable wave numbers
• generalized plane wave
• local spectral elements
• error estimates

AMS SUBJECT CLASSIFICATIONS:

• 65N30
• 65N55

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by Natural Science Foundation of China - NSF under the grant 11501529, Qinddao applied basic research project under grant 17-1-1-9-jch and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents. The second author was supported by the Natural Science Foundation of China G11571352.