An Adaptive FEM for a Maxwell Interface Problem

Duan, Huoyuan; Qiu, Fengjuan; Tan, Roger C. E.; Zheng, Weiying

doi:10.1007/s10915-015-0098-0

Cited by 17 publications

(10 citation statements)

References 43 publications

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“…For both examples, we shall first compute the numerical solutions and errors on the uniform meshes with five refinement iterations for the purpose of comparison, and then conduct the new adaptive algorithm, which is terminated when the corresponding DoFs are almost the same as that of the finest uniform mesh. The first example is modified from [22,67], where there is a corner singularity in the solution. We consider an optimal control problem on the L-shape domain:…”

Section: Numerical Experimentsmentioning

confidence: 99%

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

Zou

2021

ESAIM: M2AN

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In this work, an adaptive edge element method is developed for an H(curl)-elliptic constrained optimal control problem. We use the lowest-order Nedelec’s edge elements of first family and the piecewise (element-wise) constant functions to approximate the state and the control, respectively, and propose a new adaptive algorithm with error estimators involving both residual-type error estimators and lower-order data oscillations. By using a local regular decomposition for H(curl)-functions and the standard bubble function techniques, we derive the a posteriori error estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonal L^2- projections and the mesh-size functions to demonstrate that the sequences of the discrete states and controls generated by the adaptive algorithm converge strongly to the exact solutions of the state and control in the energy norm and L^2 -norm, respectively, by first achieving the strong convergence towards the solution to a limiting control problem. Three-dimensional numerical experiments are also presented to confirm our theoretical results and the quasi-optimality of the adaptive edge element method.

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Section: Numerical Experimentsmentioning

confidence: 99%

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

Zou

2021

ESAIM: M2AN

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“…For both examples, we shall first compute the numerical solutions and errors on the uniform meshes with five refinement iterations for the purpose of comparison, and then conduct the new adaptive algorithm, which is terminated when the corresponding DoFs are almost the same as that of the finest uniform mesh. The first example is modified from [68] [22], where there is a corner singularity in the solution. We consider an optimal control problem on the L-shape domain: 1] with both the electric permittivity σ and the magnetic permeability µ taken to be 1.…”

Section: Numerical Experimentsmentioning

confidence: 99%

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

Li¹,

Zou²

2021

Preprint

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In this work, an adaptive edge element method is developed for an H(curl)-elliptic constrained optimal control problem. We use the lowest-order Nédélec's edge elements of first family and the piecewise (element-wise) constant functions to approximate the state and the control, respectively, and propose a new adaptive algorithm with error estimators involving both residual-type error estimators and lower-order data oscillations. By using a local regular decomposition for H(curl)-functions and the standard bubble function techniques, we derive the a posteriori error estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonal L 2projections and the mesh-size functions to demonstrate that the sequences of the discrete states and controls generated by the adaptive algorithm converge strongly to the exact solutions of the state and control in the energynorm and L 2 -norm, respectively, by first achieving the strong convergence towards the solution to a limiting control problem. Three-dimensional numerical experiments are also presented to confirm our theoretical results and the quasi-optimality of the adaptive edge element method.

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“…The semidiscrete analysis for Maxwell interface problems with low regularity is provided in [43]. In addition, due to the potentially low regularity, there are many works focusing on developing a posteriori error estimators and adaptive FEM [8,16,22].…”

Section: Introductionmentioning

confidence: 99%

A Virtual Finite Element Method for Two Dimensional Maxwell Interface Problems with a Background Unfitted Mesh

Cao,

Chen,

Guo

2021

Preprint

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A virtual element method (VEM) with the first order optimal convergence order is developed for solving two dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual H(curl)-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.

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An Adaptive FEM for a Maxwell Interface Problem

Cited by 17 publications

References 43 publications

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

A Virtual Finite Element Method for Two Dimensional Maxwell Interface Problems with a Background Unfitted Mesh

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