Robust and scalable nonlinear solvers for finite element discretizations of biological transportation networks
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909568249888768 |
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| author | Haskovec, Jan Markowich, Peter Portaro, Simone Zampini, Stefano |
| author_facet | Haskovec, Jan Markowich, Peter Portaro, Simone Zampini, Stefano |
| contents | We develop robust and scalable fully implicit nonlinear finite element solvers for the simulations of biological transportation networks driven by the gradient flow minimization of a non-convex energy cost functional. Our approach employs a discontinuous space for the conductivity tensor that allows us to guarantee the preservation of its positive semi-definiteness throughout the entire minimization procedure arising from the time integration of the gradient flow dynamics using a backward Euler scheme. Extensive tests in two and three dimensions demonstrate the robustness and performance of the solver, highlight the sensitivity of the emergent network structures to mesh resolution and topology, and validate the resilience of the linear preconditioner to the ill-conditioning of the model. The implementation achieves near-optimal parallel scaling on large-scale, high-performance computing platforms. To the best of our knowledge, the network formation system has never been simulated in three dimensions before. Consequently, our three-dimensional results are the first of their kind. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Robust and scalable nonlinear solvers for finite element discretizations of biological transportation networks Haskovec, Jan Markowich, Peter Portaro, Simone Zampini, Stefano Computational Engineering, Finance, and Science 65M60, 65F08, 65Y05, 65H10 We develop robust and scalable fully implicit nonlinear finite element solvers for the simulations of biological transportation networks driven by the gradient flow minimization of a non-convex energy cost functional. Our approach employs a discontinuous space for the conductivity tensor that allows us to guarantee the preservation of its positive semi-definiteness throughout the entire minimization procedure arising from the time integration of the gradient flow dynamics using a backward Euler scheme. Extensive tests in two and three dimensions demonstrate the robustness and performance of the solver, highlight the sensitivity of the emergent network structures to mesh resolution and topology, and validate the resilience of the linear preconditioner to the ill-conditioning of the model. The implementation achieves near-optimal parallel scaling on large-scale, high-performance computing platforms. To the best of our knowledge, the network formation system has never been simulated in three dimensions before. Consequently, our three-dimensional results are the first of their kind. |
| title | Robust and scalable nonlinear solvers for finite element discretizations of biological transportation networks |
| topic | Computational Engineering, Finance, and Science 65M60, 65F08, 65Y05, 65H10 |
| url | https://arxiv.org/abs/2504.04447 |