Minimum residual discretization of a semilinear elliptic problem
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917375187615744 |
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| author | Vera, Carlos García Heuer, Norbert Praetorius, Dirk |
| author_facet | Vera, Carlos García Heuer, Norbert Praetorius, Dirk |
| contents | We propose a least-squares penalization as a means to extend the discontinuous Petrov-Galerkin (DPG) method with optimal test functions to a class of semilinear elliptic problems. The nonlinear contributions are replaced with independent unknowns so that standard DPG techniques apply to the then linear problem with non-trivial kernel. The nonlinear relations are added as least-squares constraints. Assuming solvability of the semilinear problem and an Aubin-Nitsche-type approximation property for the primal variable, we prove a Cea estimate for the approximation error in canonical norms. Numerical results with uniform and adaptively refined meshes illustrate the performance of the scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29863 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimum residual discretization of a semilinear elliptic problem Vera, Carlos García Heuer, Norbert Praetorius, Dirk Numerical Analysis 65N30, 65J15, 35J91 We propose a least-squares penalization as a means to extend the discontinuous Petrov-Galerkin (DPG) method with optimal test functions to a class of semilinear elliptic problems. The nonlinear contributions are replaced with independent unknowns so that standard DPG techniques apply to the then linear problem with non-trivial kernel. The nonlinear relations are added as least-squares constraints. Assuming solvability of the semilinear problem and an Aubin-Nitsche-type approximation property for the primal variable, we prove a Cea estimate for the approximation error in canonical norms. Numerical results with uniform and adaptively refined meshes illustrate the performance of the scheme. |
| title | Minimum residual discretization of a semilinear elliptic problem |
| topic | Numerical Analysis 65N30, 65J15, 35J91 |
| url | https://arxiv.org/abs/2603.29863 |