A Superconvergent $$C^0$$ C 0 Discontinuous Galerkin Method for Kirchhoff Plates: Error Estimates, Hybridization and Postprocessing

Abstract

This paper is devoted to a superconvergent $$C^0$$C0 discontinuous Galerkin (SCDG) method for Kirchhoff plates. First of all, following the ideas in Huang et al. (Comput Methods Appl Mech Eng 199(23---24):1446---1454, 2010) but with the normal bending moment and twisting moment as new numerical traces, we propose a modified framework of CDG methods for Kirchhoff plates. Then by a technical choice of the numerical traces, we obtain our SCDG method. Observing that the famous Hellan---Herrmann---Johnson (HHJ) method is a special case of the SCDG method, we are motivated to borrow some techniques for analyzing the HHJ method to derive optimal and superconvergent error estimates for the SCDG method. Under some assumption on the stabilization parameters, we consider the hybridization of the SCDG method. Furthermore, we construct a superconvergent discrete deflection by postprocessing the solution of the method using similar technique in Stenberg (RAIRO Model Math Anal Numer 25(1):151---167, 1991). Some numerical results are performed to demonstrate the theoretical results for the SCDG method.