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. 2019 May 21;116(21):10223-10225.
doi: 10.1073/pnas.1904139116. Epub 2019 May 6.

New Laplace and Helmholtz solvers

Abinand Gopal  1 , Lloyd N Trefethen  2
Affiliations

New Laplace and Helmholtz solvers

Abinand Gopal et al. Proc Natl Acad Sci U S A. .

Abstract

Numerical algorithms based on rational functions are introduced that solve the Laplace and Helmholtz equations on 2D domains with corners quickly and accurately, despite the corner singularities.

Keywords: Helmholtz equation; Laplace equation; rational functions; scattering.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Laplace equation in an L-shaped domain. Poles are clustered exponentially near vertices and a least-squares problem is solved on the boundary to find coefficients for a global representation [1] of the solution accurate to 10 digits. The black dot in the interior marks the point z* of [1]. The curve of maximal error on the boundary shows root-exponential convergence.
Fig. 2.
Fig. 2.
Helmholtz equation. Solutions to Δu+k2u=0 with k=50 in the exterior of a square (the real part is plotted). The incident signal on the left is a plane wave oriented at 30°, and on the right, a point oscillation H0(k|z−z0|) with z0=1/2+i.
See this image and copyright information in PMC

References

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    1. Serkh K, Rokhlin V. On the solution of elliptic partial differential equations on regions with corners. J Comput Phys. 2016;305:150–171.
    1. Wasow W. Asymptotic development of the solution of Dirichlet’s problem at analytic corners. Duke Math J. 1957;24:47–56.
    1. Newman DJ. Rational approximation to |x|. Mich Math J. 1964;11:11–14.
    1. Hochman A, Leviatan Y, White JK. On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems. J Comput Phys. 2013;238:337–358.

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