A Residual Minimization approach for Nonlinear Partial Differential Equations set in Banach spaces
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912993683439616 |
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| author | Muga, Ignacio Perera, Jorge Rojas, Sergio Ruiz-Baier, Ricardo |
| author_facet | Muga, Ignacio Perera, Jorge Rojas, Sergio Ruiz-Baier, Ricardo |
| contents | In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space and a suitably enriched discrete test space (of higher dimension than the trial space), we approximate the solution by minimizing the variational residual in a discrete dual norm. This minimization is equivalent to a nonlinear saddle-point formulation for the discrete solution in the trial space together with a residual representative in the test space. The latter provides a natural a posteriori error estimator, enabling automatic mesh adaptivity. To solve the resulting nonlinear saddle-point problem, we propose a Newton iteration whose linearized saddle-point system is symmetric, thereby guaranteeing solvability at each step. We take the $p$-Laplacian as a model problem and support the theoretical developments with representative numerical experiments, using standard $H^1$-conforming piecewise linear functions for the trial space, and lowest-order Crouzeix--Raviart functions for the test space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00341 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Residual Minimization approach for Nonlinear Partial Differential Equations set in Banach spaces Muga, Ignacio Perera, Jorge Rojas, Sergio Ruiz-Baier, Ricardo Numerical Analysis 65N30, 65N12, 65N15 In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space and a suitably enriched discrete test space (of higher dimension than the trial space), we approximate the solution by minimizing the variational residual in a discrete dual norm. This minimization is equivalent to a nonlinear saddle-point formulation for the discrete solution in the trial space together with a residual representative in the test space. The latter provides a natural a posteriori error estimator, enabling automatic mesh adaptivity. To solve the resulting nonlinear saddle-point problem, we propose a Newton iteration whose linearized saddle-point system is symmetric, thereby guaranteeing solvability at each step. We take the $p$-Laplacian as a model problem and support the theoretical developments with representative numerical experiments, using standard $H^1$-conforming piecewise linear functions for the trial space, and lowest-order Crouzeix--Raviart functions for the test space. |
| title | A Residual Minimization approach for Nonlinear Partial Differential Equations set in Banach spaces |
| topic | Numerical Analysis 65N30, 65N12, 65N15 |
| url | https://arxiv.org/abs/2604.00341 |