Error Analysis of a Conforming FEM for Multidimensional Fragmentation Equations
This work provides a rigorous framework for high-order spatial approximation in multidimensional fragmentation models, which is incremental as it extends existing methods to new dimensions.
The authors tackled the problem of solving multidimensional fragmentation equations by developing a higher-order finite element method, achieving optimal-order spatial error estimates of O(h^{r+1}) and second-order convergence in time, as validated through numerical experiments in 2D and 3D.
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.