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

Abstract

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; $$\mathbb{S}$$ S ) simultaneously with the displacement u in L2(Ω). By stemming from the structure of H(div, Ω; $$\mathbb{S}$$ S ) conforming elements for the linear elasticity problems proposed by Hu and Zhang (2014), the H(divdiv, Ω; $$\mathbb{S}$$ S ) conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H (div, Ω; $$\mathbb{S}$$ S ) conforming spaces of Pk symmetric tensors. The inheritance makes the basis functions easy to compute. The discrete spaces for u are composed of the piecewise Pk−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.