A superconvergent ultra-weak discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

This paper proposes and analyzes a superconvergent ultra-weak discontinuous Galerkin (UWDG) finite element method for nonlinear second-order two-point boundary-value problems. We first derive optimal L²-error estimates of the scheme. The order of convergence is proved to be p+ 1 in the L²-norm, when piecewise polynomials of degree p≥ 2 are used. Moreover, we prove that the UWDG solution is superconvergent with order p+ 2 for p= 2 and p+ 3 for p≥ 3 towards a special projection of the exact solution. Finally, we prove that the UWDG solution and its derivative are superconvergent at nodes with order 2p. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree p≥ 2. Numerical experiments are presented to confirm the sharpness of all the theoretical findings.