Commun. Comput. Phys., 15 (2014), pp. 1461-1479.

A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation

Lin Mu 1, Junping Wang 2, Xiu Ye 3, Shan Zhao 4*

1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
2 Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230, USA.
3 Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA.
4 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA.

Received 25 November 2012; Accepted (in revised version) 21 October 2013
Available online 12 March 2014


A weak Galerkin (WG) method is introduced and numerically tested for the Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite element formulation is symmetric and parameter free. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains. Challenging problems with high wave numbers are also examined. Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement, and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.

AMS subject classifications: 65N15, 65N30, 76D07, 35B45, 35J50

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Key words: Galerkin finite element methods, discrete gradient, Helmholtz equation, large wave numbers, weak Galerkin.

*Corresponding author.
Email: (L. Mu), (J. Wang), (X. Ye), (S. Zhao)

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