Superconvergence analysis of an ultra-weak discontinuous galerkin method for nonlinear second-order volterra integro-differential equations

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 $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norm is the use of appropriately chosen numerical fluxes together with a specially designed projection. When piecewise polynomials of degree $$p \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> are employed, the proposed UWDG method is shown to achieve the optimal convergence order of $$p+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Moreover, we prove that the UWDG solution exhibits superconvergence of order $$p+2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> toward a suitable projection of the exact solution. In addition, both the p -degree UWDG solution and its derivative are shown to be $$\mathcal {O}(h^{2p})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> superconvergent at the end of each time step. These theoretical results hold for arbitrary regular meshes and piecewise polynomial spaces of degree $$p \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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.