In this paper we develop an adaptive finite element method for elliptic problems. First, we assume that in each subdomain the norm of the approximation error at the current mesh configuration is bounded by the norm of the approximation error obtained at the previous mesh configuration, for some norm . Then an a-posteriori error estimator is designed based on the approximate solution correction between the solution on the last two mesh configurations. Based on this new error estimator, the element-wise refinement strategy in each subdomain is provided for a given tolerance.
A discussion on the choice of the coefficients in the assumption is given for different norm spaces and for different degrees of finite element family. Four 2D numerical benchmark examples of different domains and two 3D numerical benchmark examples are tested to demonstrate the robustness of our method. When possible, our numerical results are also compared to corresponding results from existing methods. All the results show that the proposed method is robust and efficient in terms of the number of degrees of freedom.
