A Simple Weak Galerkin Finite Element Method for a Class of Fourth-Order Problems in Fluorescence Tomography
Chunmei Wang, Shangyou Zhang
TL;DR
The paper addresses a class of fourth-order PDEs modeling fluorophore distributions in fluorescence tomography. It introduces a simple stabilizer-free weak Galerkin Finite Element Method based on $E_w$ and $\nabla_w$ operators and bubble-function techniques, enabling stable discretization on nonconvex polytopal meshes while preserving a symmetric positive definite structure. The authors establish optimal-order error estimates in a discrete $H^2$-norm and demonstrate sharp accuracy through numerical experiments on convex and nonconvex meshes. This work reduces implementation complexity and broadens the mesh flexibility of WG methods for FT applications.
Abstract
In this paper, we propose a simple numerical algorithm based on the weak Galerkin (WG) finite element method for a class of fourth-order problems in fluorescence tomography (FT), eliminating the need for stabilizer terms required in traditional WG methods. FT is an emerging, non-invasive 3D imaging technique that reconstructs images of fluorophore-tagged molecule distributions in vivo. By leveraging bubble functions as a key analytical tool, our method extends to both convex and non-convex elements in finite element partitions, representing a significant advancement over existing stabilizer-free WG methods. It overcomes the restrictive conditions of previous approaches, offering substantial advantages. The proposed method preserves a simple, symmetric, and positive definite structure. These advantages are confirmed by optimal-order error estimates in a discrete $H^2$ norm, demonstrating the effectiveness and accuracy of our approach. Numerical experiments further validate the efficiency and precision of the proposed method.