Pollution free discretization of the wave equation

Abstract

The Helmholtz equation governs boundary value problems in many physical applications. For high wave numbers, this equation suffers the ‘pollution effect’. The main effect of the ‘pollution’ is that the wave number of the numerical solution obtained via a finite element method is different from the wave number of the exact solution. In this article, we present three different types of discretizations, viz., ‘standard Galerkin’, ‘standard first-order system least squares (FOSLS)’ and ‘regularized FOSLS’ and numerically study this pollution effect for these three methods. Based on these discretizations, the errors are obtained in L ² and H ^1,k norms. Our numerical experiments show that the regularized FOSLS solution converges to the exact solution with O(h ²) error in the L ²-norm and with O(h) error in the H ^1,k -norm.

Keywords:

• Helmholtz equation
• Pollution
• Finite element method
• Partial differential equation

Acknowledgments

The author would like to thank Prof. Harry Yserentant, Math. Inst., Universität Tübingen, Germany, Prof. H. R. Trebin, Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Germany, Dr. Klaus Neymeyr and Dr. Gradinaru Vasile for their helpful comments and suggestions. The research contained in this paper has been supported by Deutsche Forschungsgemeinschaft as part of Sonderforschungsbereich 382.

Notes

^*c/o Dr. Sunil Goria, 1745/1 Ta Colony, Teacher Home, Pantnagar, Uttaranchal 263145, India.

Additional information

Notes on contributors

Kailash C. Patidar*

*c/o Dr. Sunil Goria, 1745/1 Ta Colony, Teacher Home, Pantnagar, Uttaranchal 263145, India.