Post-processing discontinuous Galerkin solutions to Volterra integro-differential equations: Analysis and simulations

This paper presents a superconvergence extraction technique for Volterra integro-differential equations with smooth and non-smooth kernels. Specifically, extracting superconvergence is done via a post-processed discontinuous Galerkin (DG) method obtained from interpolating the DG solution using Lagrange polynomials at the nodal points. A global superconvergence error bound (in the L"~-norm) is established. For a non-smooth kernel, a family of non-uniform time meshes is used to compensate for the singular behaviour of the exact solution near t=0. The derived theoretical results are numerically validated in a sample of test problems, demonstrating higher-than-expected convergence rates.

[1]  J. Bramble,et al.  Higher order local accuracy by averaging in the finite element method , 1977 .

[2]  Yanping Chen,et al.  Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions , 2012 .

[3]  William McLean,et al.  Discontinuous Galerkin method for an evolution equation with a memory term of positive type , 2009, Math. Comput..

[4]  Hermann Brunner.  Collocation Methods for Volterra Integral and Related Functional Differential Equations: The collocation method for ODEs: an introduction , 2004 .

[5]  Kassem Mustapha,et al.  A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels , 2013, Math. Comput..

[6]  Mehdi Dehghan,et al.  On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments , 2008, Computing.

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

[8]  William McLean,et al.  A second-order accurate numerical method for a fractional wave equation , 2006, Numerische Mathematik.

[9]  Haiying Wang,et al.  Superconvergent discontinuous Galerkin methods for second-order elliptic problems , 2009, Math. Comput..

[10]  Tao Tang,et al.  A note on collocation methods for Volterra integro-differential equations with weakly singular kernels , 1993 .

[11]  Mehdi Dehghan,et al.  The spectral methods for parabolic Volterra integro-differential equations , 2011, J. Comput. Appl. Math..

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

[13]  Hermann Brunner,et al.  Polynomial Spline Collocation Methods for Volterra Integrodifferential Equations with Weakly Singular Kernels , 1986 .

[14]  Yingjun Jiang,et al.  On spectral methods for Volterra-type integro-differential equations , 2009 .

[15]  Kassem Mustapha,et al.  A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel , 2010 .

[16]  Bernardo Cockburn,et al.  Adjoint Recovery of Superconvergent Linear Functionals from Galerkin Approximations. The One-dimensional Case , 2007, J. Sci. Comput..

[17]  Mehdi Dehghan,et al.  The numerical solution of the non-linear integro-differential equations based on the meshless method , 2012, J. Comput. Appl. Math..

[18]  Endre Süli,et al.  Enhanced accuracy by post-processing for finite element methods for hyperbolic equations , 2003, Math. Comput..

[19]  Tang,et al.  ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS , 2008 .

[20]  Jennifer K. Ryan,et al.  Smoothness-Increasing Accuracy-Conserving (SIAC) Postprocessing for Discontinuous Galerkin Solutions over Structured Triangular Meshes , 2011, SIAM J. Numer. Anal..

[21]  Jennifer K. Ryan,et al.  Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions , 2012, Math. Comput..

[22]  Hermann Brunner,et al.  Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels , 2001, SIAM J. Numer. Anal..

[23]  Hermann Brunner,et al.  The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes , 1985 .

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

[25]  William McLean,et al.  Superconvergence of a Discontinuous Galerkin Method for Fractional Diffusion and Wave Equations , 2012, SIAM J. Numer. Anal..