A reconstructed discontinuous approximation for distributed elliptic control problems Article Swipe

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YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2512.08353

In this paper, we present and analyze an internal penalty discontinuous Galerkin method for the distributed elliptic optimal control problems. It is based on a reconstructed discontinuous approximation which admits arbitrarily high-order approximation space with only one unknown per element. Applying this method, we develop a proper discretization scheme that approximates the state and adjoint variables in the approximation space. Our main contributions are twofold: (1) the derivation of both a priori and a posteriori error estimates of the $L^2$-norm and the energy norms, and (2) the implementation of an efficiently solvable discrete system, which is solved via a linearly convergent projected gradient descent method. Numerical experiments are provided to verify the convergence order in a priori estimate and the efficiency of a posteriori error estimate.

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Mathematics A priori and a posteriori Discretization Convergence (economics) Applied mathematics Approximation error Discontinuous Galerkin method Gradient descent Optimal control Galerkin method Approximation theory Space (punctuation) Mathematical analysis Scheme (mathematics) Mathematical optimization Adjoint equation Error detection and correction Approximation algorithm Descent (aeronautics) Energy (signal processing) Elliptic curve Iterative method Gradient method Discretization error A priori estimate Rate of convergence State space State (computer science)

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Type: preprint
Landing Page: https://doi.org/10.48550/arxiv.2512.08353
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OpenAlex ID: https://openalex.org/W7114766291

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DOI: https://doi.org/10.48550/arxiv.2512.08353

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Title: A reconstructed discontinuous approximation for distributed elliptic control problems

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Type: preprint

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Publication year: 2025

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Publication date: 2025-12-09

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Authors: Li, Ruo, Liu, Haoyang, Yin, Jun

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Landing page: https://doi.org/10.48550/arxiv.2512.08353

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Concepts: Mathematics, A priori and a posteriori, Discretization, Convergence (economics), Applied mathematics, Approximation error, Discontinuous Galerkin method, Gradient descent, Optimal control, Galerkin method, Approximation theory, Space (punctuation), Mathematical analysis, Scheme (mathematics), Mathematical optimization, Adjoint equation, Error detection and correction, Approximation algorithm, Descent (aeronautics), Energy (signal processing), Elliptic curve, Iterative method, Gradient method, Discretization error, A priori estimate, Rate of convergence, State space, State (computer science)

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