Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data
type: Item
instance of: scholarly article
title: Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data
zbMATH Open document ID: 0648.65078
DOI: 10.1007/BF01400356
author: Hans-Peter Helfrich
published in: Numerische Mathematik
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133213
review text: The following parabolic problem in a Hilbert space \((X,\|\|)\) is considered \((1)\quad u'(t)=-Au(t)+f(u(t)),\) \(u(0)=v\in X\) where \(A\) is linear, selfadjoint and positive definite, \(f: V\to X\) is nonlinear, satisfying the conditions \(f(0)=0\), and: \[ (2)\quad \exists a<\alpha <2,\quad \forall R>0,\quad \exists C(R)>0,\quad \forall \epsilon >0,\quad \forall u,v\quad J(\epsilon,u),J(\epsilon,v)<R,\quad \| f(u)-f(v)\| \leq C(R)\epsilon^{\alpha -}J(\epsilon,u-v) \] with \(J(\epsilon,u)=Max(\| u\|,\epsilon \| v\|_ V)\). The space \((V,\|\|_ V)\) is the domain of definition of the bilinear form \(a\), defining in the standard way the ``generalized operator \(A\)''. The first result (Theorem 1) is an existence and uniqueness theorem for the solution of (1) with the only assumption on \(v: v\in X\). Also an estimate for the solution \(u(t)\) of the following form \(\| u(t)\| +t^{1/2}\| u(t)\|_ V\leq C(v)\) \(t\in (0,t_ 0]\) is derived. Applying standard Galerkin techniques on finite element type subspaces of \(X\), a semidiscrete (dependent on \(t\)) approximation \(u_ h(t)\) is constructed for (1). Then, the author introduces certain intermediate spaces Y, \(V\subset Y\subset X\), depending on some parameter \(\beta >0\), and strengthens the condition (2) for f, making it dependent on Y. The second result (Theorem 2) are following estimates for the Galerkin error operator \(e_ h(t):\| e_ h(t)\| \leq Ch^{\mu}t^{-\mu /2}\) and \(\| e_ h(t)\|_ Y\leq Ch^{\mu}t^{-(\mu +\beta)/2}\) (\(\mu\geq 0\)). Two examples are examined.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35R20
zbMATH DE Number: 4058789
zbMATH Keywords: error estimates
zbMATH Keywords: semidiscrete Galerkin method
zbMATH Keywords: abstract semilinear evolution equations
zbMATH Keywords: optimal order of convergence
zbMATH Keywords: linear finite elements
zbMATH Keywords: Galerkin method
zbMATH Keywords: error bound
zbMATH Keywords: regularity of initial condition
MaRDI profile type: Publication
cites work: Q3027518
cites work: Quasilinear Evolution Equations and Parabolic Systems
cites work: Approximation numérique de certaines équations paraboliques non linéaires
cites work: Some Convergence Estimates for Semidiscrete Galerkin Type Approximations for Parabolic Equations
cites work: On the finite element method for parabolic equations. I: Approximation of homomorphic semi-groups
cites work: Caractérisation de quelques espaces d'interpolation
cites work: Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen
cites work: Geometric theory of semilinear parabolic equations
cites work: Finite Element Approximation of the Nonstationary Navier–Stokes Problem. I. Regularity of Solutions and Second-Order Error Estimates for Spatial Discretization
cites work: Finite Element Approximation of the Nonstationary Navier–Stokes Problem III. Smoothing Property and Higher Order Error Estimates for Spatial Discretization
cites work: Some Convergence Estimates for Semidiscrete Type Schemes for Time-Dependent Nonselfadjoint Parabolic Equations
cites work: Error Estimates for Spatially Discrete Approximations of Semilinear Parabolic Equations with Nonsmooth Initial Data
cites work: On the nonstationary Navier-Stokes systems
cites work: Defect subspaces and generalized resolvents of an Hermitian operator in the space \(\Pi_\kappa\)
cites work: Q5634306
cites work: On the Smoothing Property of the Galerkin Method for Parabolic Equations
cites work: Q3851554
cites work: Q4747774
cites work: Q5337462
cites work: Global solutions of reaction-diffusion systems
cites work: Convergence Estimates for Semidiscrete Parabolic Equation Approximations
cites work: On the rate of convergence of the difference finite element approximation for parabolic equations
cites work: Q3948553
cites work: Q4727360
cites work: Maximum-norm-stability and error-estimates in Galerkin methods for parabolic equations in one space variable
cites work: Q4169288
cites work: Q3695717
cites work: Q3886160
description: scientific article; zbMATH DE number 4058789