Superconvergence analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order volterra integro-differential equations
Year: 2026
Type: article
Abstract: This paper focuses on the analysis of convergence and superconvergence properties of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order Volterra integro-differential equations. The key idea for deriving optimal error estimates in the $$L^2$$ L 2 -norm is the use of appropriately chosen numerical fluxes together with a specially designed projection. When piecewise polynomials of degree $$p \ge 2$$ p ≥ 2 are employed, the proposed UWDG method is shown to achieve the optimal convergence order of $$p+1$$ p + 1 . Moreover, we prove that the UWDG solution exhibits superconvergence of order $$p+2$$ p + 2 toward a suitable projection of the exact solution. In addition, both the p -degree UWDG solution and its derivative are shown to be $$\mathcal {O}(h^{2p})$$ O ( h 2 p ) superconvergent at the end of each time step. These theoretical results hold for arbitrary regular meshes and piecewise polynomial spaces of degree $$p \ge 2$$ p ≥ 2 . Finally, several numerical experiments are presented to confirm the theoretical findings. Notably, the proposed UWDG method offers a significant advantage over standard discontinuous Galerkin methods for first-order systems, as it can be applied directly without introducing auxiliary variables or reformulating the problem as a larger system, thereby reducing both memory requirements and computational cost.
Source: Numerical Algorithms
Author: Mahboub Baccouch
Institution: University of Nebraska at Omaha
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W7166305490