A fully iterative adaptive energy-based approach for monotone elliptic problems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914350425440256 |
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| author | Leu, Raphael Wihler, Thomas P. |
| author_facet | Leu, Raphael Wihler, Thomas P. |
| contents | We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of adaptively refined finite-dimensional approximation spaces and employs a (nonlinear) conjugate gradient (CG) method to compute suitable approximations on each space. A core novelty of our approach is that all components of the algorithm are consistently driven by energy reduction principles rather than by classical a posteriori estimators. In particular, adaptive refinement is steered by local energy reduction indicators which aim to construct subsequent approximation spaces in a way that attains the largest potential decrease in energy. Likewise, the stopping criteria for the iterative solver are based on either relative or averaged energy reductions on each subspace. As a concrete realization, we present a concise implementation for $\mathbb{P}_1$ finite element discretizations of second-order semilinear elliptic diffusion-reaction models, where the local indicators driving the element refinements are computed based on edge-wise energy reductions. Numerical experiments demonstrate that the resulting scheme achieves optimal convergence for various benchmark problems in two-dimensional polygonal domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21913 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A fully iterative adaptive energy-based approach for monotone elliptic problems Leu, Raphael Wihler, Thomas P. Numerical Analysis 35A15, 35B38, 65J15, 65M50, 65N30 We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of adaptively refined finite-dimensional approximation spaces and employs a (nonlinear) conjugate gradient (CG) method to compute suitable approximations on each space. A core novelty of our approach is that all components of the algorithm are consistently driven by energy reduction principles rather than by classical a posteriori estimators. In particular, adaptive refinement is steered by local energy reduction indicators which aim to construct subsequent approximation spaces in a way that attains the largest potential decrease in energy. Likewise, the stopping criteria for the iterative solver are based on either relative or averaged energy reductions on each subspace. As a concrete realization, we present a concise implementation for $\mathbb{P}_1$ finite element discretizations of second-order semilinear elliptic diffusion-reaction models, where the local indicators driving the element refinements are computed based on edge-wise energy reductions. Numerical experiments demonstrate that the resulting scheme achieves optimal convergence for various benchmark problems in two-dimensional polygonal domains. |
| title | A fully iterative adaptive energy-based approach for monotone elliptic problems |
| topic | Numerical Analysis 35A15, 35B38, 65J15, 65M50, 65N30 |
| url | https://arxiv.org/abs/2602.21913 |