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⋅n and divσh⋅n, on each edge. The double divergence of the H(divdiv) mixed finite element space is exact the full discontinuous Pk polynomial space on a triangular grid. The stability of the mixed method is established. The stress solution converges at the optimal order in L2 and H(divdiv) norms. Four-order of superconvergence is proved in L2 norm for the displacement solution. The Pk discrete solution is processed on each triangle to obtain an optimal order Pk+4 solution 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+2 space while the existing mixed elements are subspaces of the H(divdiv)−Pk+2 space, on a triangular grid.
Published Version
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