Convergence analysis of a discontinuous finite element formulation based on second order derivatives

A new discontinuous Galerkin formulation is introduced for the elliptic reaction–diffusion problem that incorporates local second order distributional derivatives. The corresponding bilinear form satisfies both coercivity and continuity properties on the broken Hilbert space of H2 functions. For piecewise polynomial approximations of degree p ⩾ 2, optimal uniform h and p convergence rates are obtained in the broken H1 and H2 norms. Convergence in L2 is optimal for p ⩾ 3, if the computational mesh is strictly rectangular. If the mesh consists of skewed elements, then optimal convergence is only obtained if the corner angles satisfy a given regularity condition. For p = 2, only suboptimal h convergence rates in L2 are obtained and for linear polynomial approximations the method does not converge.