An hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems

We present an hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. We derive a priori error bound in the H1-norm that is fully explicit with respect to the local time steps and the local approximation orders. Moreover, we prove that the hp-version of the C0-continuous Petrov-Galerkin time stepping method based on geometrically refined time steps and on linearly increasing approximation orders yields exponential rates of convergence for solutions with start-up singularities. Numerical examples confirm the theoretical results.

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

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

[3]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[4]  Paul Houston,et al.  Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems , 2001, SIAM J. Numer. Anal..

[5]  Dominik Schötzau,et al.  A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations , 2010, Numerische Mathematik.

[6]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[7]  Bärbel Holm,et al.  Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up , 2014, Numerische Mathematik.

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

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

[11]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[12]  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..

[13]  B. Hulme.  Discrete Galerkin and related one-step methods for ordinary differential equations , 1972 .

[14]  Noel Walkington,et al.  Combined DG-CG Time Stepping for Wave Equations , 2014, SIAM J. Numer. Anal..

[15]  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.

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

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

[18]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[19]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[20]  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..

[21]  William W. Hager,et al.  Discontinuous Galerkin methods for ordinary differential equations , 1981 .

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

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

[24]  Tao Tang,et al.  An adaptive time stepping method with efficient error control for second-order evolution problems , 2013 .

[25]  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..

[26]  Jianguo Huang,et al.  AN ADAPTIVE LINEAR TIME STEPPING ALGORITHM FOR SECOND-ORDER LINEAR EVOLUTION PROBLEMS , 2015 .

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

[28]  Lijun Yi,et al.  An h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document}–p\documentclass[12pt]{minimal} \usepackage{amsma , 2015, Journal of Scientific Computing.

[29]  B. Hulme.  One-step piecewise polynomial Galerkin methods for initial value problems , 1972 .

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

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

[32]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[33]  D. Estep,et al.  Global error control for the continuous Galerkin finite element method for ordinary differential equations , 1994 .

[34]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[35]  Thomas P. Wihler,et al.  hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems , 2016, J. Sci. Comput..

[36]  Alfio Quarteroni,et al.  A high-order discontinuous Galerkin approximation to ordinary differential equations with applications to elastodynamics , 2018 .