Error control for the FEM approximation of an upscaled thermo-diffusion system with Smoluchowski interactions

TL;DR

This paper presents an error analysis for finite element approximations of a complex thermo-diffusion system with Smoluchowski interactions, relevant in drug delivery and contaminant transport modeling.

Contribution

It provides the first a priori error estimates for the FEM approximation of a nonlinear, nonlocal thermo-diffusion system with Smoluchowski interactions.

Findings

Error bounds for semidiscrete FEM approximations

Error bounds for fully discrete FEM schemes

Mathematical techniques include energy estimates and compactness arguments

Abstract

We analyze a coupled system of evolution equations that describes the effect of thermal gradients on the motion and deposition of $N$ populations of colloidal species diffusing and interacting together through Smoluchowski production terms. This class of systems is particularly useful in studying drug delivery, contaminant transportin complex media, as well as heat shocks thorough permeable media. The particularity lies in the modeling of the nonlinear and nonlocal coupling between diffusion and thermal conduction. We investigate the semidiscrete as well as the fully discrete em a priori error analysis of the finite elements approximation of the weak solution to a thermo-diffusion reaction system posed in a macroscopic domain. The mathematical techniques include energy-like estimates and compactness arguments.