Error Analysis of a Conforming FEM for Multidimensional Fragmentation Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908949197881344 |
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| author | Arushi Kumar, Naresh |
| author_facet | Arushi Kumar, Naresh |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_07793 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Error Analysis of a Conforming FEM for Multidimensional Fragmentation Equations Arushi Kumar, Naresh Numerical Analysis 65N15, 65N30 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. |
| title | Error Analysis of a Conforming FEM for Multidimensional Fragmentation Equations |
| topic | Numerical Analysis 65N15, 65N30 |
| url | https://arxiv.org/abs/2604.07793 |