An h-p Version of the Continuous Petrov–Galerkin Method for Nonlinear Delay Differential Equations

We investigate an h-p version of the continuous Petrov–Galerkin time stepping method for nonlinear delay differential equations with vanishing delays. We derive a priori error estimates in the $$L^{2}$$L2-, $$H^{1}$$H1- and $$L^\infty $$L∞-norm that are completely explicit with respect to the local time steps, the local polynomial degrees, and the local regularity of the exact solution. Moreover, we show that the h-p version continuous Petrov–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. The theoretical results are illustrated by some numerical experiments.

[1]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

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

[3]  Katja Bachmeier,et al.  Finite Elements Theory Fast Solvers And Applications In Solid Mechanics , 2017 .

[4]  Qiumei Huang,et al.  Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type , 2016 .

[5]  Dirk Roose,et al.  Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations , 2000, SIAM J. Sci. Comput..

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

[7]  Jie Shen,et al.  Legendre and Chebyshev dual-Petrov–Galerkin methods for Hyperbolic equations , 2007 .

[8]  Lijun Yi,et al.  An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems , 2015 .

[9]  Zhimin Zhang,et al.  The long time behavior of a spectral collocation method for delay differential equations of pantograph type , 2012 .

[10]  Chengjian Zhang,et al.  Superconvergence of a Dg Method for First-Order Linear Delay Differential Equations , 2011 .

[11]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[12]  Yunqing Huang,et al.  The Galerkin continuous finite element method for delay-differential equation with a variable term , 2007, Appl. Math. Comput..

[13]  D. Braess.  Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics , 1995 .

[14]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[15]  Li,et al.  SUPERCONVERGENCE OF A DISCONTINUOUS GALERKIN METHOD FOR FIRST-ORDER LINEAR DELAY DIFFERENTIAL EQUATIONS , 2011 .

[16]  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..

[17]  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..

[18]  Stefano Maset,et al.  Stability of Runge-Kutta methods for linear delay differential equations , 2000, Numerische Mathematik.

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

[20]  Christopher T. H. Baker,et al.  A Bibliography on the Numerical Solution of Delay Differential Equations , 2000 .

[21]  A. Manitius,et al.  A fully-discrete spectral method for delay-differential equations , 1991 .

[22]  Petra Ostermann,et al.  Collocation Methods For Volterra Integral And Related Functional Equations , 2016 .

[23]  Thomas P. Wihler,et al.  A note on a norm-preserving continuous Galerkin time stepping scheme , 2016, 1605.05201.

[24]  Hehu Xie,et al.  Discontinuous Galerkin Methods for Delay Differential Equations of Pantograph Type , 2010, SIAM J. Numer. Anal..

[25]  Jens Lorenz,et al.  An analysis of the petrov—galerkin finite element method , 1978 .

[26]  Zhongqing Wang,et al.  A LEGENDRE-GAUSS COLLOCATION METHOD FOR NONLINEAR DELAY DIFFERENTIAL EQUATIONS , 2010 .

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

[28]  Hongjiong Tian,et al.  Asymptotic Stability of Linear Neutral Delay Differential-Algebraic Equations and Runge-Kutta Methods , 2014, SIAM J. Numer. Anal..

[29]  H. Brunner,et al.  A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS * , 2009 .

[30]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

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

[32]  Marino Zennaro,et al.  Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method , 1985 .

[33]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[34]  Alexei Ilyin,et al.  One-Dimensional Interpolation Inequalities, Carlson–Landau Inequalities, and Magnetic Schrödinger Operators , 2015, 1502.01572.

[35]  Alfredo Bellen,et al.  One-step collocation for delay differential equations , 1984 .

[36]  Qiumei Huang,et al.  Local Superconvergence of Continuous Galerkin Solutions for Delay Differential Equations of Pantograph Type , 2016 .

[37]  Wang Zhong-qing,et al.  Legendre–Gauss–Lobatto spectral collocation method for nonlinear delay differential equations , 2013 .

[38]  Ivo Babuska,et al.  The p and h-p Versions of the Finite Element Method, Basic Principles and Properties , 1994, SIAM Rev..