Error Analysis of a Conforming FEM for Multidimensional Fragmentation Equations

TL;DR

This paper introduces a novel higher-order conforming finite element method for multidimensional fragmentation equations, providing rigorous analysis and validation of optimal spatial and temporal convergence rates.

Contribution

It is the first to establish a rigorous conforming finite element framework for high-order spatial approximation of multidimensional fragmentation models.

Findings

Achieves $L^2$-optimal convergence rates of ${ m O}(h^{r+1})$ in space.

Attains second-order accuracy in time with BDF2 scheme.

Numerical experiments confirm theoretical error estimates and robustness.

Abstract

In this work, we develop and analyze a higher-order finite element method for the multidimensional fragmentation equation. To the best of our knowledge, this is the first study to establish a rigorous, conforming finite element framework for high-order spatial approximation of multidimensional fragmentation models. The scheme is formulated in a variational setting, and its stability and convergence properties are derived through a detailed mathematical analysis. In particular, the $L^2$ projection operator is used to obtain optimal-order spatial error estimates under suitable regularity assumptions on the exact solution. For temporal discretization, a second-order backward differentiation formula (BDF2) is adopted, yielding a fully discrete scheme that achieves second-order convergence in time. The theoretical analysis establishes $ L^2$-optimal convergence rates of ${\cal O}(h^{r+1})$ in space, together with second-order accuracy in time. The theoretical findings are validated through a series of numerical experiments in two and three space dimensions. The computational results confirm the predicted error estimates and demonstrate the robustness of the proposed method for various choices of fragmentation kernels and selection functions.