An h–p Version of the Discontinuous Galerkin Method for Volterra Integro-Differential Equations with Vanishing Delays

We present an h–p version of the discontinuous Galerkin time stepping method for Volterra integro-differential equations with vanishing delays. We derive a priori error bounds in the $$L^2$$- and $$L^\infty $$-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Moreover, we prove that the h–p version of the discontinuous Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities. Numerical experiments are presented to illustrate the theoretical results.

[1]  Hermann Brunner,et al.  Collocation methods for nonlinear Volterra integro-differential equations with infinite delay , 1989 .

[2]  Kassem Mustapha,et al.  An hp-Version Discontinuous Galerkin Method for Integro-Differential Equations of Parabolic Type , 2011, SIAM J. Numer. Anal..

[3]  Lijun Yi,et al.  An h-p Version of the Continuous Petrov-Galerkin Finite Element Method for Volterra Integro-Differential Equations with Smooth and Nonsmooth Kernels , 2015, SIAM J. Numer. Anal..

[4]  Ishtiaq Ali,et al.  Spectral methods for pantograph-type differential and integral equations with multiple delays , 2009 .

[5]  Wayne H. Enright,et al.  Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays , 2011, SIAM J. Sci. Comput..

[6]  Tingting Meng,et al.  An h-p Version of the Continuous Petrov–Galerkin Method for Nonlinear Delay Differential Equations , 2017, Journal of Scientific Computing.

[7]  Hehu Xie,et al.  The hp Discontinuous Galerkin Method for Delay Differential Equations with Nonlinear Vanishing Delay , 2013, SIAM J. Sci. Comput..

[8]  D. Schötzau,et al.  Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations , 2014 .

[9]  Dominik Schötzau,et al.  Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method , 2000, SIAM J. Numer. Anal..

[10]  Chengjian Zhang,et al.  Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations , 2004 .

[11]  Yang Xu,et al.  Legendre Spectral Collocation Methods for Volterra Delay-Integro-Differential Equations , 2015, J. Sci. Comput..

[12]  Zhong-Qing Wang,et al.  An hp-spectral collocation method for nonlinear Volterra functional integro-differential equations with delays , 2016 .

[13]  Thomas P. Wihler,et al.  An A Priori Error Analysis of the hp-Version of the Continuous Galerkin FEM for Nonlinear Initial Value Problems , 2005, J. Sci. Comput..

[14]  Yanping Chen,et al.  Legendre Spectral Collocation Methods for Pantograph Volterra Delay-Integro-Differential Equations , 2012, J. Sci. Comput..

[15]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[16]  Jaap J. W. van der Vegt,et al.  A Space-Time Finite Element Method for Neural Field Equations with Transmission Delays , 2017, SIAM J. Sci. Comput..

[17]  Wayne H. Enright,et al.  Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay , 2012 .

[18]  Lina Wang,et al.  An h-p version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations , 2017, Advances in Computational Mathematics.

[19]  Dominik Schötzau,et al.  An hp a priori error analysis of¶the DG time-stepping method for initial value problems , 2000 .

[20]  Emiko Ishiwata.  On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays , 2000, Computing.

[21]  Stefan Vandewalle,et al.  Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays , 2009 .

[22]  Emiko Ishiwata,et al.  On the Attainable Order of Collocation Methods for Delay Differential Equations with Proportional Delay , 2000 .

[23]  W. H. Enright,et al.  Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay , 1997 .

[24]  Dominik Schötzau,et al.  hp-discontinuous Galerkin time stepping for parabolic problems , 2001 .

[25]  Dominik Schötzau,et al.  hp-Discontinuous Galerkin Time-Stepping for Volterra Integrodifferential Equations , 2006, SIAM J. Numer. Anal..