Fredrik Fryklund

Fredrik Fryklund

Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual. For the localized component, we derive explicit asymptotic expansions for volume, single layer, and double layer potentials in powers of a length scale parameter $\varepsilon$. The coefficients are expressed in terms of local geometric quantities and derivatives of the source data. The derivation is carried out entirely in the Fourier domain and applies to the Poisson equation in two and three dimensions, as well as to a class of coupled strongly elliptic systems.

Edith Frisk Gärtner, Fredrik Fryklund, Anna-Karin Tornberg

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.