Least-Squares Finite Element Methods for nonlinear problems: A unified framework
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909890961735680 |
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| author | Bertrand, Fleurianne Brodbeck, Maximilian Ricken, Tim Schneider, Henrik |
| author_facet | Bertrand, Fleurianne Brodbeck, Maximilian Ricken, Tim Schneider, Henrik |
| contents | This paper presents a unified Least-Squares framework for solving nonlinear partial differential equations by recasting the governing system as a residual minimisation problem. A Least-Squares functional is formulated and the corresponding Gauss-Newton method derived, which approximates simultaneously primal and dual variables. We derive conditions under which the Least-Squares functional is coercive and continuous in an appropriate solution space, and establish convergence results while demonstrating that the functional serves as a reliable a posteriori error estimator. This inherent error estimation property is then exploited to drive adaptive mesh refinement across a variety of problems, including the stationary heat equation with either temperature-dependent or discontinuous conductivity, nonlinear elasticity based on the Saint-Venant Kirchhoff model and sea-ice dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Least-Squares Finite Element Methods for nonlinear problems: A unified framework Bertrand, Fleurianne Brodbeck, Maximilian Ricken, Tim Schneider, Henrik Numerical Analysis This paper presents a unified Least-Squares framework for solving nonlinear partial differential equations by recasting the governing system as a residual minimisation problem. A Least-Squares functional is formulated and the corresponding Gauss-Newton method derived, which approximates simultaneously primal and dual variables. We derive conditions under which the Least-Squares functional is coercive and continuous in an appropriate solution space, and establish convergence results while demonstrating that the functional serves as a reliable a posteriori error estimator. This inherent error estimation property is then exploited to drive adaptive mesh refinement across a variety of problems, including the stationary heat equation with either temperature-dependent or discontinuous conductivity, nonlinear elasticity based on the Saint-Venant Kirchhoff model and sea-ice dynamics. |
| title | Least-Squares Finite Element Methods for nonlinear problems: A unified framework |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.18739 |