A Posteriori Error Estimation for the Discontinuous Galerkin Method Applied to Nonlinear Volterra Integro-differential Equations

Abstract In this paper, we present and rigorously analyze a novel residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method applied to nonlinear Volterra integro-differential equations. We first prove that the DG solution using piecewise polynomials of degree $$p \geqslant 1$$ <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> achieves the optimal convergence rate of $$\mathcal {O}(h^{p+1})$$ <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:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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. Moreover, we show that the DG solution exhibits $$\mathcal {O}(h^{2p+1})$$ <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:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> superconvergence at downwind points. These results are used to show that the DG solution is $$\mathcal {O}(h^{p+2})$$ <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:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> superconvergent towards a specific Gauss-Radau projection of the exact solution. This superconvergence result enables a decomposition of the discretization error into two components: a dominant term proportional to a right Radau polynomial of degree $$(p+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and a remainder that converges at a rate of $$\mathcal {O}(h^{p+2})$$ <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:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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. The global superconvergence properties are used to construct a residual-based a posteriori error estimator. The proposed estimator is computationally efficient, easy to implement, and asymptotically exact. It is constructed by solving local residual problems without imposing initial conditions on each element. We establish convergence of the a posteriori error estimator to the true error at the optimal rate of $$\mathcal {O}(h^{p+2})$$ <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:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> under mesh refinement. Finally, we prove that the global effectivity index of the estimator approaches unity at a rate of $$\mathcal {O}(h)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Several numerical examples are provided to validate the theoretical results and to illustrate the accuracy and reliability of the proposed estimator.