A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems
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 is investigated for an isotropic discretization of a problem with anisotropic diffusion but without convection, this paper proposed residual error estimator that allows one to prove an optimal lower bound. The factor \(\varepsilon^{-1/2}\) in the upper bound in above mentioned paper is retained in authors analysis, since the alignment measure is of the same order in the isotropic case. The presented experiments show, however, that the effectivity index is bounded uniformly in \(\varepsilon\) on adequately refined anisotropic meshes. The authors rigorously analyze a new a posteriori error estimate for the finite element approximation of anisotropic - convection - reaction equations with anisotropic finite elements. The equivalence between the energy norm of the error and the residual error estimator is proved. It is proved that this estimate is reliable and efficient (numerical experiments confirm proposed theoretical predictions).
streamline upwind Petrov-Galerkin method
scientific article; zbMATH DE number 5863178
scientific article