Lightning Algorithm for Inhomogeneous Modified Helmholtz Equations on Cornered Domains and Their Applications

Abstract: We develop in this paper an efficient solver for the inhomogeneous modified Helmholtz equations in two-dimensional complex domains with particular attention to resolving corner singularities. We represent the solution as a summation of a particular solution and the solution of the modified Helmholtz equation. Built upon suitable domain embedding methods, we compute the particular solution using the Fourier spectral method and/or Legendre--Fourier spectral method. To treat the corner singularities, we develop a lightning algorithm by employing an expansion of the modified Bessel functions of the second kind, coupled with tapered exponentially clustering poles, to approximate the corner singularities, and using an expansion of the modified Bessel functions of the first kind to approximate the smooth part of the solution. Finally, we determine the unknown expansion coefficients by solving a least-squares problem to satisfy the boundary conditions. Numerical experiments demonstrate that root-exponential convergence can be achieved for the modified Helmholtz equation with corner singularities. To further demonstrate the applicability of the proposed algorithm, it has been successfully applied to solve some typical linear and nonlinear parabolic equations. We have also made a first attempt in extending the lightning algorithm to a three-dimensional modified Helmholtz equation.