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

Abstract: 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.

• The paper introduces a Fourier-based framework that eliminates the need for explicit Green's functions by leveraging the Fourier symbol and parabolic regularization.
• It derives explicit asymptotic expansions for localized potential components in both 2D and 3D Poisson problems as well as strongly elliptic systems.
• The method supports high-order, fast computational solvers adaptable to complex geometries, achieving convergence rates consistent with theoretical predictions.

Fourier-Based Potential Theory Without an Explicit Green's Function

Introduction and Motivation

Fourier-based solution methods and classical potential theory are foundational to the computation of solutions for linear constant-coefficient PDEs. Traditional potential theory techniques are reliant on the explicit availability of Green's functions in physical space to represent solutions in terms of single and double layer potentials. This dependence forms a key limitation: for many coupled systems of equations, complex domains, and multiphysics contexts, closed-form Green's functions are unavailable, restricting the utility of integral equation approaches.

The presented work introduces a general framework for Fourier-based potential theory that eliminates the necessity of an explicit Green's function. The central innovation is to leverage the Fourier symbol of the governing operator to construct a parabolic regularization, yielding a decomposition into a nonlocal, mollified component and a spatially localized, exponentially decaying residual. Through a fully Fourier domain procedure, explicit asymptotic expansions are constructed for the localized part, with coefficients determined by local geometry and derivatives of source data. This approach is systematically applied to the Poisson equation in two- and three-dimensional Euclidean spaces as well as to strongly elliptic coupled systems, significantly generalizing the domain of potential-theoretic fast solvers.

Theoretical Framework

Parabolic Regularization and Fourier Symbol Approach

The proposed method replaces the Green's function by operating directly with the Fourier symbol $M(\xi)$ of the elliptic operator. For the Poisson equation, $M(\xi) = |\xi|^2$, and the solution is retrieved from the Fourier domain inversion, regularized via a parabolic factor (Gaussian convolution in physical space).

The solution $u(x)$ to $-\Delta u(x) = g(x)$ is represented as a sum: $u(x) = u_H(x) + L(x)$ with

$$u_H(x) = \mathcal{F}^{-1}\left[ \frac{e^{-|\xi|^2 \varepsilon}}{|\xi|^2} \hat{g}(\xi) \right] \quad \text{and} \quad L(x) = \mathcal{F}^{-1}\left[ \frac{1 - e^{-|\xi|^2 \varepsilon}}{|\xi|^2} \hat{g}(\xi) \right]$$

Here, $u_H$ is a smooth, nonlocal term suitable for evaluation by FFT or its non-uniform and adaptive variants, while $L$ is sharply localized and admits systematic asymptotic expansion in small $\varepsilon$.

For more general systems

$$M(\xi) = \text{symbol matrix of a strongly elliptic system}$$

the heat kernel decomposition generalizes to

$M(\xi) = |\xi|^2$0

and the regularized splitting applies correspondingly. The only requirement is invertibility and positive real part of the eigenvalues.

Asymptotic Expansion of the Localized Component

A hallmark of the framework is the explicit derivation of asymptotic expansions for localized components (layer and volume potentials) as power series in $M(\xi) = |\xi|^2$1. The methodology works entirely in the Fourier domain, relying on differentiation and Taylor expansions of the symbol and the source density. The expansions take the form: $M(\xi) = |\xi|^2$2 where $M(\xi) = |\xi|^2$3 involve geometric invariants (curvature, higher derivatives at the projection point on the boundary, surface Laplacian, etc.) and derivatives of the data (density or source function).

For sources exhibiting discontinuities or singular support (e.g., layer densities on a boundary curve), the expansion naturally incorporates fractional powers of $M(\xi) = |\xi|^2$4 (e.g., half-integers). Derivations use tools from the theory of tempered distributions to handle distributional sources, ensuring rigorous treatment of nearly singular integrals.

Generalization to Complex Domains and Coupled Systems

The spatial localization and structure of the expansions depend solely on the operator symbol, requiring only customary smoothness and invertibility conditions. As such, the approach can be extended to:

• Poisson equations in both $M(\xi) = |\xi|^2$5D and $M(\xi) = |\xi|^2$6D, yielding explicit expansions for volume, single layer, and double layer potentials that recover all known Green’s function-based results.
• Strongly elliptic coupled systems (including first-order and zeroth-order couplings), where the matrix symbol may not be diagonalizable by constant coefficients and explicit Green’s functions are not generally available.

The coupling structure appears in the asymptotics through terms in the expansion of matrix exponentials, inducing cross-correlation between system components at higher order.

Numerical Implementation and Performance

The theoretical framework is explicitly amenable to fast computational methods:

• The smooth, nonlocal components $M(\xi) = |\xi|^2$7 are computed efficiently with FFT or kernel-independent fast algorithms, including adaptive and non-uniform FFTs or DMK [dmk2024].
• The asymptotic expansions of localized corrections $M(\xi) = |\xi|^2$8 are evaluated at each target point via a finite number of local geometric and derivative evaluations, sidestepping singular quadrature and geometrically complex integration.

The work demonstrates, through numerical experiments employing manufactured solutions on domains with nontrivial geometry (e.g., tori), that the order of convergence matches the predicted asymptotic rate, i.e., errors obey $M(\xi) = |\xi|^2$9 for truncation order $u(x)$0 in the expansion. Convergence persists even for target points within vanishingly small distance to the boundary, directly addressing one of the main obstacles for volume and layer potential evaluation in traditional methods.

Implications and Prospects

This approach fundamentally changes the practice of potential theory in several aspects:

• Elimination of Green's function restrictions: Applicability to PDEs and coupled systems without available explicit Green's function, addressing a primary bottleneck for diverse multiphysics applications.
• Local adaptivity and high-order accuracy: The parabolic regularization and associated expansions permit high-order, locally refined evaluation—critical for domains with non-trivial topology or locally high curvature.
• Fast solvers in general geometries: The method is synergistic with recent advances in hierarchical and dual-space kernel-splitting algorithms [dmk2024], enabling efficient large-scale simulations.
• Theoretical generality: The framework extends to matrix-valued elliptic operators, distributions as source terms, and potentially to scenarios where only the Fourier response is empirically accessible (data-driven systems).

Future work may focus on integration with adaptive discretization, parabolic (time-dependent) extensions, and leveraging data-driven estimation of the operator symbol in inverse problems or learning contexts.

Conclusion

The presented Fourier-based potential theory framework provides a robust, generalizable, and computationally effective approach to the evaluation of volume and layer potentials for elliptic PDEs and coupled systems without resort to explicit Green's functions. The derivations, conducted entirely in the Fourier domain, yield explicit, geometry-aware asymptotic expansions for the local, singular corrections, and support the construction of high-order, adaptive fast solvers in arbitrary geometries. This work opens new avenues for the treatment of PDEs in complex and coupled settings and is poised to have broad implications for computational PDE, numerical analysis of integral equations, and scientific computing.

References:

• "Fourier-based potential theory without an explicit Green's function" (2604.11436)
• "A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials" [lghtwght_ps2026]
• "A Dual-Space Multilevel Kernel-Splitting Framework for Discrete and Continuous Convolution" [dmk2024]