Adaptive Approximations of Inclusions in a Semilinear Elliptic Problem Related to Cardiac Electrophysiology
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911093500149760 |
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| author | Jin, Bangti Wang, Fengru Xu, Yifeng |
| author_facet | Jin, Bangti Wang, Fengru Xu, Yifeng |
| contents | In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite element method for the resulting constrained minimization problem that is relaxed by a phase-field approach. The \textit{a posteriori} error estimators of the adaptive algorithm consist of three components, i.e., the state variable, the adjoint variable and the complementary relation. Moreover, using tools from adaptive finite element analysis and nonlinear optimization, we establish the strong convergence for a subsequence of adaptively generated discrete solutions to a solution of the continuous optimality system. Several numerical examples are presented to illustrate the convergence and efficiency of the adaptive algorithm |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Approximations of Inclusions in a Semilinear Elliptic Problem Related to Cardiac Electrophysiology Jin, Bangti Wang, Fengru Xu, Yifeng Numerical Analysis Optimization and Control In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite element method for the resulting constrained minimization problem that is relaxed by a phase-field approach. The \textit{a posteriori} error estimators of the adaptive algorithm consist of three components, i.e., the state variable, the adjoint variable and the complementary relation. Moreover, using tools from adaptive finite element analysis and nonlinear optimization, we establish the strong convergence for a subsequence of adaptively generated discrete solutions to a solution of the continuous optimality system. Several numerical examples are presented to illustrate the convergence and efficiency of the adaptive algorithm |
| title | Adaptive Approximations of Inclusions in a Semilinear Elliptic Problem Related to Cardiac Electrophysiology |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2504.04483 |