Abstract
A reliable and computable a posteriori error bound is derived for the mixed Ciarlet–Raviart method for the equation \(\Delta\Delta u+\kappa^{2}u=f(x)\) , \(x\in\Omega\subset R^{2}\) , with the first boundary condition and a piecewise constant \(\kappa\geq 0\) . Several authors derived residual type a posteriori error bounds at assumptions that \(\kappa\equiv 0\) and the domain is polygonal, none of which is used in the paper. In case of a piecewise smooth boundary, we consider the mixed method with the triangular Lagrange finite elements of the \(3^{\textrm{d}}\) order, which, in general, are curvilinear along the boundary. This provides an approximation to the boundary, matching the finite elements in accuracy. Our bounds belong to the class of a posteriori functional majorants and are evaluated with help of functions from the testing \(C^{1}\) space. By the reasons of accuracy and simplicity, this space is generated by the finite elements with the domains, coinciding with the domains of the Lagrange elements, and singular rational coordinate functions.
