Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes

Abstract

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous Pk polynomials are employed. To connect discontinuous functions, the inter-element traces, {uh} and {∇uh}, are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted Pk+4 polynomial from four Pk polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the Pk solution has an O(hk+5) convergence in L2-norm, and an O(hk+3) convergence in H2-norm. A local post-process is defined, which lifts the Pk solution to a Pk+4 quasi-optimal solution. Numerical tests are provided, confirming the theory.