A Superconvergent Local Discontinuous Galerkin Method for Elliptic Problems

In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show ( (+1)) convergence rates for the solution and its gradient and ( (+2)) superconvergent solutions at Radau points on enriched -degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a -degree discontinuous finite element solution is spanned by two (+1)-degree right Radau polynomials in the and directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is ( (+2)) superconvergent at tensor product of the roots of left Radau polynomial in and right Radau polynomial in while the second component is ( (+2)) superconvergent at the tensor product of the roots of the right Radau polynomial in and left Radau polynomial in . Several numerical simulations are performed to validate the theory.