Proximal Discontinuous Galerkin Methods for Variational Inequalities
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917429106442240 |
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| author | Ern, Alexandre Keith, Brendan Kim, Dohyun Masri, Rami Riviere, Beatrice |
| author_facet | Ern, Alexandre Keith, Brendan Kim, Dohyun Masri, Rami Riviere, Beatrice |
| contents | We introduce a family of proximal discontinuous Galerkin methods for variational inequalities, focusing on the obstacle problem as a didactic example. Each member of this family is born from applying a different well-known nonconforming finite element discretization to the Bregman proximal point method. We explicitly treat four examples: the symmetric interior penalty discontinuous Galerkin, the enriched Galerkin, the hybridizable interior penalty and the hybrid high-order methods. We formulate a unified analysis framework for this family of methods and prove the existence and uniqueness of solutions, energy dissipation, and error estimates for both the primal and dual variables. Remarkably, the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine facet unknowns leads to the first higher-order convergence result for any proximal Galerkin method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19708 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proximal Discontinuous Galerkin Methods for Variational Inequalities Ern, Alexandre Keith, Brendan Kim, Dohyun Masri, Rami Riviere, Beatrice Numerical Analysis 35J86, 49J40, 65N30 We introduce a family of proximal discontinuous Galerkin methods for variational inequalities, focusing on the obstacle problem as a didactic example. Each member of this family is born from applying a different well-known nonconforming finite element discretization to the Bregman proximal point method. We explicitly treat four examples: the symmetric interior penalty discontinuous Galerkin, the enriched Galerkin, the hybridizable interior penalty and the hybrid high-order methods. We formulate a unified analysis framework for this family of methods and prove the existence and uniqueness of solutions, energy dissipation, and error estimates for both the primal and dual variables. Remarkably, the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine facet unknowns leads to the first higher-order convergence result for any proximal Galerkin method. |
| title | Proximal Discontinuous Galerkin Methods for Variational Inequalities |
| topic | Numerical Analysis 35J86, 49J40, 65N30 |
| url | https://arxiv.org/abs/2604.19708 |