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 {greater than or slanted equal to} 2, optimal uniform h and p convergence rates are obtained in the broken H¹ and H² norms. Convergence in L² is optimal for p {greater than or slanted equal to} 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.