Fast unified evaluation of layer and volume potentials for the 2D modified Helmholtz equation
For computational scientists solving boundary value problems, this method offers a unified, efficient, and robust framework for potential evaluations, though it is an incremental improvement over existing fast multipole and NUFFT-based approaches.
The paper presents a fast and accurate unified method for evaluating layer and volume potentials for the 2D modified Helmholtz equation, achieving high throughput and robustness to mesh quality without stabilization techniques.
We present a fast and accurate potential theory-based method for the two-dimensional modified Helmholtz equation, treating the involved singular and nearly singular layer evaluations together with volume potentials within a single computational framework. The method is based on a decomposition of the free-space Green's function into a short-range local part and a smooth long-range part. The long-range contribution is evaluated efficiently using the non-uniform fast Fourier transform (NUFFT), while the local contribution is treated by asymptotic expansions. For the layer potentials, an intermediate telescoping sum over dyadic refinement levels is added, where the resulting difference kernels are smooth and rapidly decaying, allowing the dyadic levels to be evaluated without specialized quadrature rules. The volume potential is evaluated on triangular cut-cell meshes, where the mesh only enters the scheme as quadrature rule for smooth data. This makes the method robust with respect to small and distorted mesh cells, without the need for stabilization or cell-merging techniques. Numerical experiments demonstrate the expected convergence rates, high throughput of the potential evaluations, and robustness with respect to mesh quality.