Solving Laplace and Helmholtz problems with corner singularities via rational functions and their analogs
It is well known that the solutions of elliptic partial differential equations on domains with corners typically develop singularities. As such, a straightforward application of numerical methods developed for smooth problems will often yield prohibitively slow convergence. Numerous techniques for overcoming this difficulty when using finite elements and boundary integral equations have been proposed over the years, but they can often be challenging to implement in practice. We take a significantly different approach motivated by a result of D. J. Newman in 1964 on the rational approximation of |x|. The resulting method is capable of solving the planar Laplace and low-frequency Helmholtz equations on domains with several corners to high accuracy in under a second. Moreover, the method can be implemented in just a few dozen lines of code. This is joint work with Nick Trefethen.
Abi Gopal is a Ph.D. student in the Numerical Analysis Group at the University of Oxford, advised by Prof. Nick Trefethen. Prior to this, he completed his undergraduate degree in applied mathematics at Virginia Tech. His doctoral work has primarily focused on high-order solvers for partial differential equations, specifically, methods based on integral equations and approximation theory, but he is broadly interested in numerical analysis, numerical linear algebra, and approximation theory.