A family of H-div-div mixed triangular finite elements for the biharmonic equation

Abstract

A family of Pk+22×2-PkH(divdiv)mixed finite elements on triangular grids is constructed for the biharmonic equation. For the H(divdiv)mixed finite element functions, both the stress tensor and its divergence have continuous normal components, σh⋅nand divσh⋅n, on each edge. The double divergence of the H(divdiv)mixed finite element space is exact the full discontinuous Pkpolynomial space on a triangular grid. The stability of the mixed method is established. The stress solution converges at the optimal order in L2and H(divdiv)norms. Four-order of superconvergence is proved in L2norm for the displacement solution. The Pkdiscrete solution is processed on each triangle to obtain an optimal order Pk+4solution there. Numerical results are presented verifying the theory. Additional computation shows that all existing H(divdiv)mixed finite elements fail to solve jump-coefficient biharmonic equations while the new mixed element converges independently of the interface-jump. This is because the H(divdiv)element here is the full H(divdiv)−Pk+2space while the existing mixed elements are subspaces of the H(divdiv)−Pk+2space, on a triangular grid.

Keywords

• Mixed finite element
• Div-div finite element
• Biharmonic equation
• Triangular mesh