Goal-oriented error estimation for transient parabolic problems
type: Item
instance of: scholarly article
title: Goal-oriented error estimation for transient parabolic problems
zbMATH Open document ID: 1172.65051
author: P. Diez
author: Giovanni Calderón
published in: Computational Mechanics
publication date: 2009-05-08T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/8479
review text: The paper focuses on the estimation of errors present when solving transient parabolic problems with finite element methods (FEMs). The authors consider the parabolic equation \(\dot u-\nabla \cdot (\alpha \nabla u)=f\) in \( \Omega \times I\) where \(\Omega \) is a bounded domain of \(\mathbb{R}^{d}\), \( d=1,2,3\), and \(I=(0,T)\). An initial data \(u(x,0)=u_{0}(x)\) is given and homogeneous Dirichlet boundary conditions are imposed on \(\partial \Omega \times I\). When applying a FEM discretization, a weak formulation of the problem is written as: find \(u^{H}\in \mathcal{V}^{H}\) such that \((\dot{u}^{H}(t),v)+a(u^{H}(t),v)=l(v)\) for every \(v\in \mathcal{V}^{H}\), where \( \mathcal{V}^{H}\) is a subspace of \(\mathcal{H}_{0}^{1}(\Omega )\). A time discretization is introduced through a time grid decomposing \(I\) in subintervals \(I_{n}\) and considering a space \(\mathcal{V}_{\Delta t}=\{v:I\rightarrow \mathbb{R};v\mid _{I_{n}}\in \mathcal{P}_{q}(I_{n})\}\), where \(\mathcal{P}_{q}(I_{n})\) is the space of polynomials of degree less than \(q\) in \(I_{n}\). A discontinuous Galerkin procedure is then used which combines \(\mathcal{V}^{H}\) and \(\mathcal{V}_{\Delta t}\) in \(\mathcal{V} _{\Delta t}^{H}\) and consists to write a weak formulation of the problem in the space \(\mathcal{V}_{\Delta t}^{H}\): find \(u^{H}\in \mathcal{V}_{\Delta t}^{H}\) such that \(B(u^{H},v)=L(v)\) for every \(v\in \mathcal{V}_{\Delta t}^{H}\). This weak formulation involves the jump of \(u^{H}\) at \(t^{n}\). In order to compute the errors of some quantities called of interest, the authors write the weak formulation of the original problem: find \(u\in \mathcal{V}\) such that \(B(u,v)=L(v)\) for every \(v\in \mathcal{V}\). The authors consider quantities of the kind \(J(u)=(j(u(.,T)))\). They introduce a dual problem \(B(v,\varphi )=J(v)\) for every \(v\in \mathcal{V}\) and derive the problem whose \(\varphi \) is a solution. The idea of the paper consists first to insert the erro…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5552314
zbMATH Keywords: transient parabolic problem
zbMATH Keywords: finite element method
zbMATH Keywords: discontinuous Galerkin discretization
zbMATH Keywords: weak formulation
zbMATH Keywords: error estimation
zbMATH Keywords: numerical examples
Wikidata QID: Q57547836
MaRDI profile type: Publication
OpenAlex ID: W2084200726
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DOI: 10.1007/S00466-006-0106-1
description: scientific article; zbMATH DE number 5552314