Asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method
The author analyzes an a posteriori error estimate based on the equilibrated residual method, for the elliptic boundary value problem \((\Omega \subset\mathbb{R}^2\) is a bounded polygon with boundary \(\partial \Omega = \Gamma_D \cup \Gamma_N\), \(\text{Int} (\Gamma_D)\cap\text{Int} (\Gamma_N)=\emptyset\) and \(c \geq 0)\) \[ -\triangle u + cu = f\text{ in }\Omega\;\partial u | \partial n = g\text{ on }\Gamma_N, u=0\text{ on }\Gamma_D. \tag{1} \] The sole intention of the author is to prove a desirable property (i.e. asymptotical exactness) for an a-posteriori estimator. The equilibrated residual method is introduced. Main result: A local superconvergence is proved for the case of mixed boundary conditions. As the solution \(u\) of the variational form of the problem (1) belongs to the space \(H^3(\Omega_1) \cap H^2_\infty (\Omega_1) (\Omega_0 \subset \subset \Omega_1 \subset \subset \Omega\) and \(\Omega_1 \subset \Omega_{1h}\) for \(h\) small enough and the mesh is quasi-uniform on \( \Omega_1 \), \(h\)= max diam\((K)\) where \(K\) are the triangles), \(K \in \mathcal{T} _h\), the following estimate holds: \[ \left(\sum _{K \in \mathcal{T}_{h} (\Omega_0)}| | | e-\Phi_K | | | _K^2\right)^{1/2} \leq c(\Omega_0 , \Omega_1) \{h^{1+\rho}(| | u| | _{3,2,\Omega_1}+ | u| _{2,\infty, \Omega_1}) + | | e| | _{0,2,\Omega_1}\}, \] with \(\rho = \)min\((\alpha, {1\over2}, {\sigma\over2})\) (here any node of an element intersecting \(\Omega_0\) has exactly 6 triangles attached to it and \(\Phi_K \in V_K = \{ v \in H^1 (K): v =0 \) on \( \Gamma_D \cap \partial K \}\) is the solution of the local residual problem: \[ B_K(\Phi_K,v)=(f,\nu)_K - B_K(u_h, v)+ \int_{\partial K} g_K \;v\, ds,\;\forall v \in V_K \] where \(B_K (u,v)= \int_K (\nabla u . \nabla v +c \;u \;v)\,dx\), and \(e=u-u_h\) denotes the error of the approximation). Some numerical experiments that illustrate the conclusions of the main Theorem are presented. Finally, a result about the structure \(O(h^{1+\alpha})\) (an a posteriori error estimator is asymptotically exact in the energy norm for regular solutions and \(0(h^{1+\alpha}) (\alpha >O)\) meshes) is proved.
scientific article; zbMATH DE number 5151310