A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs

Abstract

A novel method for solving boundary value problems for linear partial differential equations in convex polygons was developed by A.S. Fokas in the late 1990s. A spectrally accurate numerical implementation for the case of the Laplace equation led the way for the present implementation for both the Helmholtz equation and the modified Helmholtz equation. We obtain again spectrally accurate numerical solutions for Dirichlet to Neumann maps, without any discretization of the domain interior. The eigenvalue problem is also examined using the present Legendre expansion method. A key feature of the current approach is that all integrals that arise in the map problem can be evaluated in closed form, reducing it to the solution of a small overdetermined linear system of equations.

Keywords:

• Fokas transform method
• boundary integral method
• Dirichlet– to–Neumann map
• Helmholtz-type equations
• polygonal domain
• Legendre polynomials

AMS Subject Classifications:

• 65E05
• 65R10
• 35C15
• 31A10
• 35J25

Acknowledgments

Discussions with A.S. Fokas and N. Flyer are gratefully acknowledged. The work by B. Fornberg was supported by the NSF grant DMS-0914647.