A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems

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• 作者：Chunguang Xiong ; Roland Becker ; Fusheng Luo and Xiuling Ma
• 刊名：Numerical Methods for Partial Differential Equations
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：33
• 期：1
• 页码：318-353
• 全文大小：2090K
• ISSN：1098-2426

文摘

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three-dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable p = − Δ u , we rewrite the problem into a two-order system. Then, the a priori error estimates are derived in L² norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r ( r ≥ 1 ) are used, we obtain the optimal convergence rate of order r + 1 in L² norm and of order r in DG norm for u, and the order r in both norms for p = − Δ u . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies.