A Weak Galerkin Finite Element Method for A Type of Fourth Order Problem Arising From Fluorescence Tomography

TL;DR

This paper introduces a weak Galerkin finite element method tailored for solving a specific fourth order problem from fluorescence tomography, demonstrating optimal error estimates and efficiency through numerical experiments.

Contribution

It develops a novel weak Galerkin finite element approach for a fourth order problem in fluorescence tomography, with proven optimal error estimates and practical numerical validation.

Findings

Optimal order error estimates in $H^2$-norm

Optimal convergence in $L^2$ norm for most schemes

Numerical experiments confirm efficiency and accuracy

Abstract

In this paper, a new and efficient numerical algorithm by using weak Galerkin (WG) finite element methods is proposed for a type of fourth order problem arising from fluorescence tomography(FT). Fluorescence tomography is an emerging, in vivo non-invasive 3-D imaging technique which reconstructs images that characterize the distribution of molecules that are tagged by fluorophores. Weak second order elliptic operator and its discrete version are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. An error estimate of optimal order is derived in an $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.