A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids
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This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress σ=-∇^2u is sought in the Sobolev space H(divdiv,Ω;𝕊) simultaneously with the displacement u in L^2(Ω). Stemming from the structure of H(div,Ω;𝕊) conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the H(divdiv,Ω;𝕊) conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div,Ω;𝕊) conforming spaces of P_k symmetric tensors. The inheritance makes the basis functions easy to compute. The discrete spaces for u are composed of the piecewise P_k-2 polynomials without requiring any continuity. Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k≥ 3, and the optimal order of convergence is achieved. Besides, the superconvergence and the postprocessing results are displayed. Some numerical experiments are provided to demonstrate the theoretical analysis.
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