1. An hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem.
- Author
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Feng, Jinhua, Wang, Shixi, Bi, Hai, and Yang, Yidu
- Subjects
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GALERKIN methods , *MATHEMATICAL physics , *FINITE element method , *BIHARMONIC equations , *EIGENFUNCTIONS , *EIGENVALUES - Abstract
• For the biharmonic eigenvalue problem, Wang et al. [Adv. Comput. Math. 45(2019)] first analyzed the mixed DG method (MDG method) under the condition that the eigenfunction u and Δ u. have the same regularity. In this paper, we establish an MDG discretization which is different with that in the existing work and we conduct our analysis under the condition that u and Δ u have different regularities. • The existing literature derived the error estimate of biharmonic eigenvalue by using the estimate of Laplacian eigenvalue problem. In this paper, we establish an identity (see Lemma 3.5) and use it to get the a priori error estimate of biharmonic eigenvalue. We prove that the convergence rate of approximate eigenvalues reaches 2p-2 in h and p − (2 s − 7) in when p ≥ 2 and the eigenfunction u ∈ H S (Ω) (s ≥ p + 1) which is consistent with the estimate in the existing work [Babuska and Osborn, Eigenvalue problems, 1991] for the classical mixed finite element method for biharmonic eigenvalue problem. • Using the identity (see Lemma 3.5) we also discuss the a posteriori error estimate for the biharmonic eigenvalue. The literature on the a posteriori error estimate for Ciarlet-Raviart mixed method is still quite a few. We give an a posteriori error estimator of the approximate eigenvalue and analyze its reliability. The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p ≥ 3 , we derive the a priori error estimates of the approximate eigenfunction in the broken H 1 norm and L 2 norm which are optimal in h and suboptimal in p. When p = 2 , the approximate eigenfunctions converge but with only suboptimal convergence rate. When p ≥ 2 and the eigenfunction u ∈ H s (Ω) (s ≥ p + 1), we prove that the convergence rate of approximate eigenvalues reaches 2 p − 2 in h and p − (2 s − 7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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