A robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the L2-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

In this paper, we investigate the Galerkin spectral approximation for elliptic control problems with integral control and state constraints. Firstly, an a posteriori error estimator is established,which can be acted as the equivalent indicatorwith explicit expression. Secondly, appropriate base functions of the discrete spacesmake it is probable to solve the discrete system. Numerical test indicates the reliability and efficiency of the estimator, and shows the proposed method is competitive for this class of control problems. These discussions can certainly be extended to two- and three-dimensional cases.

In this paper, a new a posteriori error estimator for nonconforming convection diffusionapproximation problem, which relies on the small discrete problems solution in stars, hasbeen established. It is equivalent to the energy error up to data oscillation without anysaturation assumption nor comparison with residual estimator

In this paper we develop a residual based a posteriori error analysis for an augmentedmixed finite element method applied to the problem of linear elasticity in the plane.More precisely, we derive a reliable and efficient a posteriori error estimator for thecase of pure Dirichlet boundary conditions. In addition, several numericalexperiments confirming the theoretical properties of the estimator, andillustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are also reported.