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Wang, Lixiu; Zhang, Qian; Sun, Jiguang; Zhang, Zhimin

A priori and a posteriori error estimates for the quad-curl eigenvalue problem. (English) Zbl 1490.65256

ESAIM, Math. Model. Numer. Anal. 56, No. 3, 1027-1051 (2022).
Summary: In this paper, we consider a priori and a posteriori error estimates of the \(H(\mathrm{curl}^2)\)-conforming finite element when solving the quad-curl eigenvalue problem. An a priori estimate of eigenvalues with convergence order \(2(s-1)\) is obtained if the corresponding eigenvector \(u\in H^{s-1}(\Omega)\) and \(\nabla\times u\in H^{s}(\Omega)\). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the errors of eigenvectors in the energy norm and upper bounds for the errors of eigenvalues. Numerical examples are presented for validation.

Cited in 3 Documents

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics

Keywords:

quad-curl problem; eigenvalue problem; a priori error estimation; a posteriori error estimation; curl-curl conforming elements
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References:

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