A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids

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 stressσ=-▽~2u is sought in the Sobolev space H(divdiv,Ω;S) simultaneously with the displacement u in L~2(Ω).By stemming from the structure of H(div,Ω;S) conforming elements for the linear elasticity problems proposed by Hu and Zhang (2014),the H(divdiv,Ω;S) conforming finite element spaces are constructed by imposing the normal continuity of divσon the H(div,Ω;S) conforming spaces of Pksymmetric tensors.The inheritance makes the basis functions easy to compute.The discrete spaces for u are composed of the piecewise Ppolynomials 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.