Hidden Accuracy and Superconvergence Analysis of Central Discontinuous Galerkin Methods on Overlapping Meshes

Abstract

This paper establishes the first rigorous superconvergence theory for semidiscrete and fully discrete central discontinuous Galerkin (CDG) methods for linear hyperbolic equations on overlapping meshes. While the optimal $L^2$ convergence of $\mathbb{Q}^k$ CDG schemes was established on uniform Cartesian meshes by Liu, Shu, and Zhang [ SIAM J. Numer. Anal.}, 56 (2018), pp. 520--541], their observed $\mathcal{O}(h^{k+2})$ pointwise superconvergence has remained unproven, due to the loss of standard single-mesh Galerkin orthogonality inherent in the CDG overlapping structure. To overcome this fundamental barrier, we introduce a projection-correction framework that identifies a hidden superconvergent mechanism: an asymptotic weak residual cancellation in one dimension, and a high-order cancellation-by-aggregation (HOCA) mechanism in multiple dimensions. This HOCA approach overcomes the analytical challenge posed by coupled primal-dual directional residuals, recovering critical error cancellation properties absent from the standard variational formulation. Consequently, we provide the rigorous proof of the conjectured $\mathcal{O}(h^{k+2})$ pointwise superconvergence in the discrete $\ell^{\infty}$ norm across all superconvergent points. Furthermore, we reveal that under a systematically corrected initialization, this framework yields a previously undiscovered, stronger cell-average superconvergence estimate of order $\mathcal{O}(h^{\min\{2k+1,k+3\}})$. The theory is extended to fully discrete explicit Runge--Kutta CDG schemes, where stagewise corrected errors are constructed to preserve spatial superconvergence up to temporal truncation errors, yielding a stable reconstruction-based postprocessing estimate. Numerical experiments in one and two spatial dimensions confirm the sharpness of the theoretical rates.