Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods
type: Item
instance of: scholarly article
title: Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods
zbMATH Open document ID: 1188.65108
author: Ricardo H. Nochetto
author: G. Akrivis
author: Charalambos Makridakis
published in: Numerische Mathematik
publication date: 2010-02-24T00:00:00Z
review text: The main purpose of the paper is to introduce a new methodology for performing a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge-Kutta collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ordinary differential equations and parabolic partial differential equations. The first section is an introduction in nature. The second section presents an abstract framework for the discretization and time reconstruction, with emphasis on the simple linear case. These results are applied to cG and then extended to nonlinear equations in the third section. The fourth section deals with RK-C. The authors discuss the relation between the classical order and the stage order. Thus, viewing RK-C methods as collocation or Galerkin-type methods, they clarify the connection between the stage order and the order of convergence in \(L^\infty([0,T];H)\). The latter is \(O(k^{q+1})\) because the approximate solution is a piecewise polynomial of degree \(q\). In the last section one provides explicit conditions on the data and the reconstruction which guarantee the validity for time discrete case.
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65L05
zbMATH DE Number: 5674531
zbMATH Keywords: Runge-Kutta Collocation methods
zbMATH Keywords: continuous Galerkin method
zbMATH Keywords: stiff ODE, parabolic PDE
zbMATH Keywords: global error bound
zbMATH Keywords: a posteriori error bound
zbMATH Keywords: optimal order
zbMATH Keywords: time stepping methods
zbMATH Keywords: convergence
reviewed by: Romulus Militaru
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: s00211-009-0254-2
OpenAlex ID: W2157441081
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DOI: 10.1007/S00211-009-0254-2
description: scientific article; zbMATH DE number 5674531