On the accuracy of a local-structure-preserving LDG method for the Laplace equation

In this paper, the accuracy is further examined for a local-structure-preserving local discontinuous Galerkin method, originally proposed in [F. Li and C.-W. Shu, A local-structurepreserving local discontinuous Galerkin method for the Laplace equation, Methods and Applications of Analysis, v13 (2006), pp.215-233] for solving the Laplace equation. With its distinctive feature in using harmonic polynomials as local approximations, this method has lower computational complexity than standard discontinuous Galerkin methods. Based on the primal formulation of the bilinear form of the scheme, the missing optimal L error estimate is established in this paper together with an optimal error estimate in a new energy norm. In addition, with a local post-processing procedure, it is demonstrated numerically that this cost efficient method has some accuracy loss when measured in negative-order norms.