Galerkin/Runge-Kutta discretizations of nonlinear parabolic equations
type: Item
instance of: scholarly article
title: Galerkin/Runge-Kutta discretizations of nonlinear parabolic equations
zbMATH Open document ID: 1220.65123
author: Eskil Hansen
published in: Journal of Computational and Applied Mathematics
publication date: 2007-06-14T00:00:00Z
review text: Numerical analysis of the nonlinear parabolic equation of the type \[ u_t=\nabla \cdot F(u) \;\;\text{ in } \Omega \times (0,T], \] \[ u=0, \;\;\text{ on } \partial \Omega \times (0,T],\;\;u(0) = \eta \;\text{ in } \Omega \] is investigated. The main aim is to derive a more general convergence analysis valid for higher-order methods as well as for a large family of maps \(F\) (including the \(p\) -Laplacian with \(p\geq 2\)). A full Galerkin/Runge-Kutta discretization of this nonlinear problem is used. For space discretization standard convergence analysis of nonlinear elliptic equations is used where the monotonicity properties of \(F\) are supposed. For time discretization the B-convergence theory of stiff ODEs and its generalization in author's previous paper is extended to an infinite dimensional setting. For this full discrete numerical scheme a global error bounds are derived in \(L_2\) by \(\Delta x^{r/2} +\Delta t^q\), where \(r\) is the convergence order of the Galerkin method applied to the underlying stationary problem and \(q\) is the stiff order of the algebraically stable Runge-Kutta method. One space dimensional example concludes the paper and confirms the convergence properties of the proposed method.
reviewed by: Angela Handlovičová
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65M15
zbMATH DE Number: 5164878
zbMATH Keywords: nonlinear parabolic equations
zbMATH Keywords: Galerkin/Runge-Kutta methods
zbMATH Keywords: logarithmic Lipschitz constants
zbMATH Keywords: B-convergence
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: j.cam.2006.03.041
OpenAlex ID: W2036235982
cites work: Q4094029
cites work: Finite Element Approximation of the Parabolic p-Laplacian
cites work: Stability Criteria for Implicit Runge–Kutta Methods
cites work: Finite element error estimates for nonlinear elliptic equations of monotone type
cites work: Numerical approximation of generalized Newtonian fluids using Powell-Sabin-Heindl elements: I. theoretical estimates
cites work: Q3890207
cites work: Q3776932
cites work: Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces
cites work: Sur la B-stabilité des méthodes de Runge-Kutta
cites work: Q3687624
cites work: The Concept of B-Convergence
cites work: Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems
cites work: Q4714145
cites work: Runge-Kutta time discretizations of nonlinear dissipative evolution equations
cites work: Logarithmic norms and nonlinear DAE stability
cites work: Numerical Analysis of Parabolic p-Laplacian: Approximation of Trajectories
cites work: On the numerical solution of a class of nonlinear parabolic problems with Volterra operators by a Rothe-Galerkin finite element method
cites work: Runge-Kutta Approximation of Quasi-Linear Parabolic Equations
cites work: Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems
cites work: Stability and convergence at the PDE/stiff ODE interface
cites work: Bounds on nonlinear operators in finite-dimensional Banach spaces
cites work: Q4360234
cites work: Convergence of method of lines approximations to partial differential equations
cites work: Existence, Uniqueness, and Numerical Analysis of Solutions of a Quasilinear Parabolic Problem
cites work: Q3995594
DOI: 10.1016/J.CAM.2006.03.041
description: scientific article; zbMATH DE number 5164878