Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

[1]  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..

[2]  A. Stuart,et al.  On the computation of blow-up , 1990, European Journal of Applied Mathematics.

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

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

[5]  Charalambos Makridakis,et al.  Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations , 2004, Math. Comput..

[6]  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..

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

[8]  D. Estep.  A posteriori error bounds and global error control for approximation of ordinary differential equations , 1995 .

[9]  Charalambos Makridakis,et al.  A space-time finite element method for the nonlinear Schröinger equation: the discontinuous Galerkin method , 1998, Math. Comput..

[10]  Claes Johnson.  Error Estimates and Adaptive Time-Step Control for a Class of One-Step Methods for Stiff Ordinary Differential Equations , 1988 .

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

[12]  Tomoyasu Nakagawa,et al.  Blowing up of a finite difference solution tout = uxx + u2 , 1975 .

[13]  Ricardo H. Nochetto,et al.  A posteriori error analysis for higher order dissipative methods for evolution problems , 2006, Numerische Mathematik.

[14]  Charalambos Makridakis,et al.  Galerkin time-stepping methods for nonlinear parabolic equations , 2004 .

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

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

[17]  T. Sideris.  Ordinary Differential Equations and Dynamical Systems , 2013 .

[18]  François Dubeau,et al.  Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations , 1986 .

[19]  Ricardo G. Durán,et al.  An Adaptive Time Step Procedure for a Parabolic Problem with Blow-up , 2002, Computing.

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

[21]  Endre Süli,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[22]  Ricardo H. Nochetto,et al.  Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods , 2009, Numerische Mathematik.

[23]  H. Brunner,et al.  Numerical analysis of semilinear parabolic problems with blow-up solutions. , 1994 .

[24]  Charalambos Makridakis,et al.  A Space-Time Finite Element Method for the Nonlinear Schrödinger Equation: The Continuous Galerkin Method , 1999 .