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A C0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H2-norm and the L2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence. [ABSTRACT FROM AUTHOR]

A conforming discontinuous Galerkin (CDG) C 0 - P k finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C 0 - P k finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C 0 - P k finite elements. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established. Numerical results are presented to verify the theory. [ABSTRACT FROM AUTHOR]

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous P k polynomials are employed. To connect discontinuous functions, the inter-element traces, { u h } and { ∇ u h } , 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 P k + 4 polynomial from four P k 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 P k solution has an O (h k + 5) convergence in L 2 -norm, and an O (h k + 3) convergence in H 2 -norm. A local post-process is defined, which lifts the P k solution to a P k + 4 quasi-optimal solution. Numerical tests are provided, confirming the theory. [ABSTRACT FROM AUTHOR]