PPubMed6 May 2019Cited 41×

New Laplace and Helmholtz solvers

Abinand Gopal, Lloyd N. Trefethen

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.

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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.

What’s new

i.Introduced numerical algorithms based on rational functions that solve the Laplace and Helmholtz equations on 2D domains with corners.
ii.Achieved root‑exponential convergence by prescribing exponentially clustered poles outside the domain and solving a linear least‑squares problem.
iii.Demonstrated 10‑digit accuracy with about N≈1,000 degrees of freedom and evaluation times of only tens of microseconds per point.
iv.Extended the approach to Helmholtz problems, obtaining four‑digit accuracy for scattering examples with modest computational effort.

How they did it

Poles of a rational function are placed a priori outside the domain with exponential clustering near each vertex; the unknown coefficients are then obtained by least‑squares fitting on boundary sample points, leading to a linear system of size roughly 3N×N. The resulting representation u(z)=Re r(z) (or its Helmholtz analogue) provides a global solution that can be evaluated anywhere with microsecond cost.

What they found

10‑digit accuracy achieved for an L‑shaped Laplace problem using N=1,002 degrees of freedom.
Coefficient computation takes less than 1 second on a laptop; each point evaluation takes a few tens of microseconds.
Helmholtz scattering problems solved to four‑digit accuracy in about 2 seconds, with ~250 µs per evaluation.

Caveats

✕Limited to small or medium‑size two‑dimensional problems and to constant‑coefficient PDEs with known special solutions.
✕Ill‑conditioning of the linear systems remains a challenge and is not yet fully understood.
✕Root‑exponential convergence makes the method less competitive for very high accuracies (e.g., >15 digits).

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