Numerical solution for an aggregation equation with degenerate diffusion

TL;DR

This paper introduces a stabilized finite element numerical method with semi-implicit Euler time integration for solving an aggregation equation with degenerate diffusion, proving convergence to the unique weak solution.

Contribution

It develops a novel stabilized finite element approach with rigorous convergence analysis for a nonlinear aggregation equation with degenerate diffusion.

Findings

Method converges to the unique weak solution.

Nonnegativity is maintained due to stabilization.

Numerical example demonstrates effectiveness.

Abstract

A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra stabilizing term plus a semi--implicit Euler time integration. Then we carry out a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that the sequence of finite element approximations converges toward the unique weak solution of the model at hands. In doing so, nonnegativity is attained due to the stabilizing term and the acuteness on partitions of the computational domain, and hence a priori energy estimates of finite element approximations are established. As we deal with a nonlinear problem, some form of strong convergence is required. The key compactness result is obtained via an adaptation of a Riesz--Fr\'echet--Kolmogorov criterion by perturbation. A numerical example is also presented.