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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.15351 |
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| _version_ | 1866918162457427968 |
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| author | Chen, Long Guo, Ruchi Wei, Jingrong Zou, Jun |
| author_facet | Chen, Long Guo, Ruchi Wei, Jingrong Zou, Jun |
| contents | A DualTPD method is proposed for solving nonlinear partial differential equations. The method is characterized by three main features. First, decoupling via Fenchel--Rockafellar duality is achieved, so that nonlinear terms are discretized by discontinuous finite element spaces, yielding block-diagonal mass matrices and closed-form updates. Second, improved convergence is obtained by applying transformed primal--dual (TPD) dynamics to the nonlinear saddle-point system, which yields strongly monotone behavior. Third, efficient preconditioners are designed for the elliptic-type Schur complement arising from the separated differential operators, and multigrid solvers are applied effectively. Extensive numerical experiments on elliptic $p$-Laplacian and nonlinear $H(\curl)$ problems are presented, showing significant efficiency gains with global, mesh-independent convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15351 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Novel Preconditioning Framework for Solving Nonlinear PDEs based on Fenchel-Rockafellar Duality and Transformed Primal-Dual Techniques Chen, Long Guo, Ruchi Wei, Jingrong Zou, Jun Numerical Analysis 65Y20, 65N12, 49N15 A DualTPD method is proposed for solving nonlinear partial differential equations. The method is characterized by three main features. First, decoupling via Fenchel--Rockafellar duality is achieved, so that nonlinear terms are discretized by discontinuous finite element spaces, yielding block-diagonal mass matrices and closed-form updates. Second, improved convergence is obtained by applying transformed primal--dual (TPD) dynamics to the nonlinear saddle-point system, which yields strongly monotone behavior. Third, efficient preconditioners are designed for the elliptic-type Schur complement arising from the separated differential operators, and multigrid solvers are applied effectively. Extensive numerical experiments on elliptic $p$-Laplacian and nonlinear $H(\curl)$ problems are presented, showing significant efficiency gains with global, mesh-independent convergence. |
| title | A Novel Preconditioning Framework for Solving Nonlinear PDEs based on Fenchel-Rockafellar Duality and Transformed Primal-Dual Techniques |
| topic | Numerical Analysis 65Y20, 65N12, 49N15 |
| url | https://arxiv.org/abs/2510.15351 |