Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations

TL;DR

This paper introduces a novel high-order exponentially fitted finite element method for multi-dimensional drift-diffusion equations, enhancing numerical stability and efficiency in biomolecular electrodiffusion modeling.

Contribution

It develops a new approach to construct high-order exponentially fitted basis functions using divergence-free spaces, with proven first-order convergence for specific finite element spaces.

Findings

Constructed high-order 2-D exponentially fitted basis functions.

Proved first-order convergence for divergence-free Raviart-Thomas space.

Demonstrated potential for improved numerical stability and efficiency.

Abstract

A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-one correspondence between continuous piecewise polynomial space of degree $k+1$ and the divergence-free vector space of degree $k$, one can construct high-order 2-D exponentially fitted basis functions that are strictly interpolative at a selected node set but are discontinuous on edges in general, spanning nonconforming finite element spaces. First order convergence was proved for the methods constructed from divergence-free Raviart-Thomas space $RT_0^0$ at two different node sets