A parametrix method for elliptic surface PDEs

TL;DR

This work develops a parametrix-based integral equation framework for elliptic PDEs posed on smooth surfaces $\Gamma$ in $\mathbb{R}^3$, transforming variable-coefficient surface problems into second-kind Fredholm equations using an approximate Green's function. The authors construct a Laplace–Beltrami parametrix from the planar Green's function, extend the approach to general elliptic surface PDEs with or without advection, and address boundary value problems on open surfaces. They present a Nyström discretization with specialized quadratures (singular, nearly singular, smooth) and a fast IFMM-based solver to achieve high-order accuracy and scalable performance on general geometries. Numerical experiments on spheres, ellipsoids, and a wavy torus demonstrate robust convergence for LB, Helmholtz–Beltrami, and variable-coefficient problems, including boundary-value cases. The framework offers a versatile, well-conditioned alternative to direct discretizations for complex surface PDEs with potential extensions to curves and anisotropic operators.

Abstract

Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries.

Figures (13)

• Figure 1: This plot shows the remainder kernel $R_{\text{LB}}(\boldsymbol{x},\cdot)$ for the Laplace-Beltrami problem on an example surface. We see that it is bounded, but not smooth around $\boldsymbol{x}$ (red dot).
• Figure 2: The figure shows the remainder kernel for an example surface PDE $c(x,y,z)=2x$ and $a=1$. We see that the remainder function has a singularity at points where $c(\boldsymbol{x})=0$ (the left images) and is a bounded function of $\boldsymbol{x}$ (the right images). It is also clear that the remainder inherits the character of the PDE: it is oscillatory when $c>0$ and rapidly decays when $c<0$.
• Figure 3: This figure shows an example of an open surface $\Gamma$, which is a subset of the ellipsoid $\tilde{\Gamma}$ with two patches removed. It also shows the binormal vector $\boldsymbol\nu$, which is normal to the $\partial\Gamma$, but tangent to $\Gamma$.
• Figure 4: The results of a convergence test for singular quadrature and adaptive integration routines applied to functions of the form $F(r,\theta)\sigma(x,y)$ for different choices of $F$ and $\sigma$. Both tests involve repeating the experiment for a number of different target locations and reporting the maximum error observed over all test points. The figures demonstrates that both routines can be used to integrate functions with the (near-) singularities present in $\mathcal{R}$ and $\mathcal{K}$ to the desired tolerance.
• Figure 5: The results of increasing $k_{s}$ in our IFMM. We see that for $k_{s}=10,000$, the IFMM has the same accuracy as the deterministic IFMM, but only involves a $17\%$ of the points. This fact allows us to compute the IFMM to the same tolerance in a fraction of the time.

Theorems & Definitions (20)

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