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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 O(h p+1 ) L 2 convergence rates for the solution and its gradient and O(h p+2 ) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p + 1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p + 1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h p+2 ) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h p+2 ) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

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We analyze the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the second-order wave equation in one space dimension. With a suitable projection of the initial conditions for the LDG scheme, we prove that the LDG solution and its spatial derivative are O(h p+3/2 ) super close to particular projections of the exact solutions for pth-degree polynomial spaces. We use these results to show that the significant parts of the discretization errors for the LDG solution and its derivative are proportional to (p + 1)-degree right and left Radau polynomials, respectively. These results allow us to prove that the p-degree LDG solution and its derivative are O(h p+3/2 ) superconvergent at the roots of (p + 1)-degree right and left Radau polynomials, respectively, while computational results show higher O(h p+2 ) convergence rate. Superconvergence results can be used to construct asymptotically correct a posteriori error estimates by solving a local steady problem on each element. This will be discussed further in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivative converge to the true spatial errors in the L 2 -norm under mesh refinement.