arXiv subjects

Fredrik Fryklund

Publications and source records attributed to Fredrik Fryklund.

11 recordsLinked to original sources

arXiv2026

Fast unified evaluation of layer and volume potentials for the 2D modified Helmholtz equation

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.

math.NA↗

arXiv2026

Fourier-based potential theory without an explicit Green's function

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.

math.AP↗

arXiv2025

Scattering theory for Stokes flow in complex branched structures

Slow, viscous flow in branched structures arises in many biological and engineering settings. Direct numerical simulation of flow in such complicated multi-scale geometry, however, is a computationally intensive task. We propose a scattering theory framework that dramatically reduces this cost by decomposing networks into components connected by short straight channels. Exploiting the phenomenon of rapid return to Poiseuille flow (Saint-Venant's principle in the context of elasticity), we compute a high-order accurate scattering matrix for each component via boundary integral equations. These precomputed components can then be assembled into arbitrary branched structures, and the precomputed local solutions on each component can be assembled into an accurate global solution. The method is modular, has negligible cost, and appears to be the first full-fidelity solver that makes use of the return to Poiseuille flow phenomenon. In our two-dimensional examples, it matches the accuracy of full-domain solvers while requiring only a fraction of the computational effort.

math.NA↗

arXiv2024

A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials

Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for the source distribution within the domain, followed by solving a boundary integral equation to impose the specified boundary conditions. Here, we present a new fast algorithm which is easy to implement and compatible with virtually any discretization technique, including unstructured domain triangulations, such as those used in standard finite element or finite volume methods. Our approach combines earlier work on potential theory for the heat equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT), and the dual-space multilevel kernel-splitting (DMK) framework. It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various other degeneracies. On a single CPU core, the scheme computes the solution at a rate comparable to that of the fast Fourier transform (FFT) in work per gridpoint.

math.NA↗

arXiv2023

An accurate and efficient scheme for function extensions on smooth domains

A new scheme is proposed to construct an n-times differentiable function extension of an n-times differentiable function defined on a smooth domain D in d-dimensions. The extension scheme relies on an explicit formula consisting of a linear combination of n+1 function values in D, which extends the function along directions normal to the boundary. Smoothness tangent to the boundary is automatic. The performance of the scheme is illustrated by using function extension as a step in a numerical solver for the inhomogeneous Poisson equation on multiply connected domains with complex geometry in two and three dimensions. We show that the modest additional work needed to do function extension leads to considerably more accurate solutions of the partial differential equation.

math.NA↗

arXiv2022

An FMM Accelerated Poisson Solver for Complicated Geometries in the Plane using Function Extension

We describe a new, adaptive solver for the two-dimensional Poisson equation in complicated geometries. Using classical potential theory, we represent the solution as the sum of a volume potential and a double layer potential. Rather than evaluating the volume potential over the given domain, we first extend the source data to a geometrically simpler region with high order accuracy. This allows us to accelerate the evaluation of the volume potential using an efficient, geometry-unaware fast multipole-based algorithm. To impose the desired boundary condition, it remains only to solve the Laplace equation with suitably modified boundary data. This is accomplished with existing fast and accurate boundary integral methods. The novelty of our solver is the scheme used for creating the source extension, assuming it is provided on an adaptive quad-tree. For leaf boxes intersected by the boundary, we construct a universal "stencil" and require that the data be provided at the subset of those points that lie within the domain interior. This universality permits us to precompute and store an interpolation matrix which is used to extrapolate the source data to an extended set of leaf nodes with full tensor-product grids on each. We demonstrate the method's speed, robustness and high-order convergence with several examples, including domains with piecewise smooth boundaries.

math.NA↗

arXiv2022

An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

Boundary integral methods are attractive for solving homogeneous linear constant coefficient elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy

math.NA↗

arXiv2022

An adaptive kernel-split quadrature method for parameter-dependent layer potentials

Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, biharmonic and Stokes equations. These equations depend on a parameter, denoted $α$, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter $α$, at an increased cost that scales as $\logα$. Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible.

math.NA↗

arXiv2021

An adaptive kernel-split quadrature method for parameter-dependent layer potentials

Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, modified biharmonic and modified Stokes equations. These equations depend on a parameter, denoted $α$, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter $α$, at an increased cost that scales as $\log α$. Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible.

math.NA↗

arXiv2019

An integral equation based numerical method for the forced heat equation on complex domains

Integral equation based numerical methods are directly applicable to homogeneous elliptic PDEs, and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, extensions to problems with inhomogeneous source terms and time dependent PDEs, such as the heat equation, have been introduced. One such approach for the heat equation is to first discretise in time, and in each time-step solve a so-called modified Helmholtz equation with a parameter depending on the time step size. The modified Helmholtz equation is then split into two parts: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

math.NA↗

arXiv2017

Partition of Unity Extension of Functions on Complex Domains

We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in $2D$. Function extension plays a fundamental role in extending the applicability of boundary integral methods to inhomogeneous partial differential equations with embedded domain techniques. Overlapping partitions are placed along the boundaries, and a local extension of the function is computed on each patch using smooth radial basis functions; a trivially parallel process. A partition of unity method blends the local extrapolations into a global one, where weight functions impose compact support. The regularity of the extended function can be controlled by the construction of the partition of unity function. We evaluate the performance of the PUX method in the context of solving the Poisson equation on multiply connected domains using a boundary integral method and a spectral solver. With a suitable choice of parameters the error converges as a tenth order method down to $10^{-14}$.

math.NA↗