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A new numerical approach to the solution of the 2-D Helmholtz equation with optimal accuracy on irregular domains and Cartesian meshes

  • A. IdesmanEmail author
  • B. Dey
Original Paper
  • 43 Downloads

Abstract

A new numerical approach for the time independent Helmholtz equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. At similar 9-point stencils, the accuracy of the new approach is two orders higher for the Dirichlet boundary conditions and one order higher for the Neumann boundary conditions than that for the linear finite elements. The numerical results for irregular domains also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. The new approach can be equally applied to the Helmholtz and screened Poisson equations.

Keywords

Helmholtz equation Local truncation error Irregular domains Cartesian meshes Optimal accuracy 

Notes

Acknowledgements

The research has been supported in part by the NSF Grant CMMI-1935452 and by Texas Tech University.

Supplementary material

466_2020_1814_MOESM1_ESM.pdf (26 kb)
Supplementary material 1 (pdf 26 KB)
466_2020_1814_MOESM2_ESM.pdf (38 kb)
Supplementary material 2 (pdf 37 KB)
466_2020_1814_MOESM3_ESM.pdf (46 kb)
Supplementary material 3 (pdf 45 KB)

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© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTexas Tech UniversityLubbockUSA

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