Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods.

[1]  Ricardo H. Nochetto,et al.  Pointwise a posteriori error estimates for monotone semi-linear equations , 2006, Numerische Mathematik.

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

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

[4]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems IV: nonlinear problems , 1995 .

[5]  Thomas J. R. Hughes,et al.  Encyclopedia of computational mechanics , 2004 .

[6]  Kenneth Eriksson,et al.  Adaptive Computational Methods for Parabolic Problems , 2004 .

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

[8]  Kenneth Eriksson,et al.  Adaptive Finite Element Methods for Parabolic Problems VI: Analytic Semigroups , 1998 .

[9]  Michel Crouzeix,et al.  Linearly implicit methods for nonlinear parabolic equations , 2003, Math. Comput..

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

[11]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

[12]  V. Thomée,et al.  Single step methods for inhomogeneous linear differential equations in Banach space , 1982 .

[13]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[14]  G. Wanner,et al.  Perturbed collocation and Runge-Kutta methods , 1981 .

[15]  Alexander Ostermann,et al.  RUNGE-KUTTA METHODS FOR PARABOLIC EQUATIONS AND CONVOLUTION QUADRATURE , 1993 .

[16]  Marco Picasso,et al.  An Anisotropic Error Estimator for the Crank--Nicolson Method: Application to a Parabolic Problem , 2009, SIAM J. Sci. Comput..

[17]  Peter Monk,et al.  Continuous finite elements in space and time for the heat equation , 1989 .

[18]  Ricardo H. Nochetto,et al.  A posteriori error estimates for the Crank-Nicolson method for parabolic equations , 2005, Math. Comput..

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

[20]  C. Lubich,et al.  Runge-Kutta approximation of quasi-linear parabolic equations , 1995 .

[21]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

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

[23]  Roy D. Williams,et al.  Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations , 2000 .