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.
- a priori error estimate
- Second-order boundary-value problems
- Ultra-weak discontinuous Galerkin method
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics