A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems
type: Item
instance of: scholarly article
title: A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems
zbMATH Open document ID: 1210.65187
author: Thomas Apel
author: Serge Nicaise
author: Dieter Sirch
published in: Journal of Computational and Applied Mathematics
publication date: 2011-03-09T00:00:00Z
review text: This paper is devoted to the singularly perturbed diffusion-convection-reaction problem with a special focus on anisotropic diffusion. This problem is defined by: Find the solution \( u \in H^1 (\Omega)\) of \[ \begin{cases} - \text{div} (A \nabla u) + \mathbf{b}. \nabla u + cu = f\text{ in } \Omega\\ u=0\text{ on } \Gamma_D\\ A \nabla u . \mathbf{n} = g\text{ on }\Gamma_N \end{cases} \tag{1} \] with \(f \in L_2(\Omega)\) and \(g \in L^2 (\Gamma_N), \Omega \subset \mathbb R^d, d = 2 \text{ or }3\), is a bounded domain with a polygonal \((d=2)\) of polyhedral \((d=3)\) boundary \(\Gamma = \Gamma_D \cup \Gamma_N\) (\(\Gamma_D\) and \(\Gamma_N\) are disjoint and open in \(\Gamma\)-Dirichlet and Neumann boundary conditions). Here \(\mathbf{A}\) becomes small in some direction \[ \mathbf{A} = \left( \begin{matrix} \varepsilon & 0 \\ 0 & 1 \\ \end{matrix} \right) , \;d = 2 \quad\text{ or }\quad \mathbf{A} = \left( \begin{matrix} \varepsilon & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{matrix} \right),\;d =3, \] \(\mathbf{b} \in W^{1, \infty} (\Omega)^d, c \in L^\infty (\Omega), \exists c_o \geq 0: c -1/2 \text{ div } \mathbf{b} \geq c_o\), \(\mathbf{b}. \mathbf{n} \geq 0 \) on \(\Gamma_N\), \(\mathbf{A} \in \mathbb R^{d \times d}\) is symmetric and there exists \(\alpha_\diamond > 0: \mathbf{A}\xi . \xi \geq \alpha_\diamond\), for any \(\mathbf{\xi} \in \mathbb R^D, \Gamma_D = \emptyset\) if \(c_\diamond = 0\). The authors develop an estimator of residual type for problems with convection, reaction and anisotropic diffusion. For the discretization, the \(h\)-version of the streamline upwind Petrov-Galerkin method is used. Without the stabilization term, the method reduces to a standard Galerkin method and produces non-physical oscillations. We also point out that authors error estimator works as well in this case. In comparison with the paper \textit{F. Fierro} and \textit{A. Veeser} [Numer. Math. 103, No. 2, 267--298 (2006; Zbl 1101.65104)], where a posteriori error estimation…
reviewed by: Jan Lovíšek
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N15
zbMATH DE Number: 5863178
zbMATH Keywords: anisotropic diffusion
zbMATH Keywords: SUPG
zbMATH Keywords: a posteriori error estimate
zbMATH Keywords: anisotropic finite elements
zbMATH Keywords: elliptic problem
zbMATH Keywords: optimal lower bound
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: bubble functions
zbMATH Keywords: interpolation
zbMATH Keywords: seminorm
zbMATH Keywords: diffusion-convection-reaction problem
zbMATH Keywords: streamline upwind Petrov-Galerkin method
zbMATH Keywords: stabilization
MaRDI profile type: Publication
full work available at URL: j.cam.2010.11.032
OpenAlex ID: W2055792370
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DOI: 10.1016/J.CAM.2010.11.032
description: scientific article; zbMATH DE number 5863178