Efficient boundary elements for the Smoluchowski diffusion equation

TL;DR

This paper introduces efficient boundary element methods in the frequency domain for solving the Smoluchowski diffusion equation, crucial for modeling diffusion with external forces in soft matter physics.

Contribution

The paper develops highly accurate boundary element methods that combine Fourier integral approximation with singularity resolution for the Smoluchowski equation.

Findings

Methods demonstrate high accuracy in numerical experiments.

Approach is efficient for computing rheological quantities.

Applicable to boundary value problems with unbounded coefficients.

Abstract

The Smoluchowski diffusion equation describes diffusion in the presence of external forces. Studying the mechanical response of soft materials to linear forces, such as shear, results in a boundary value problem involving an Ornstein-Uhlenbeck operator in an exterior domain with non-constant, unbounded coefficients. In this article, we present efficient and highly accurate boundary element methods in the frequency domain, motivated by applications in soft matter physics. Our key contributions concern the accurate assembly of the Galerkin matrix, combining the approximation of the fundamental solution as a Fourier integral with the resolution of near-field singularities. Numerical experiments demonstrate the accuracy and efficiency of the proposed methods and show their relevance for the computation of rheological quantities.