Convergence and optimality of adaptive nonconforming finite element methods for nonsymmetric and indefinite problems
- Recently an adaptive nonconforming finite element method (ANFEM) has been developed by Carstensen and Hoppe. In this paper, we extend their result to nonsymmetric and indefinite problems. The main tools in our analysis are a posteriori error estimators and a quasi-orthogonality property. In this case, we need to overcome two main difficulties: one stems from the nonconformity of the finite element space, the other is how to handle the effect of a nonsymmetric and indefinite bilinear form. Two ANFEM algorithms (ANFEM I, ANFEM II) are proposed for the lowest order Crouzeix-Raviart element. It is shown that both ANFEM algorithms are a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops. Moreover, optimality in the sense of optimal algorithmic complexity can be shown for ANFEM II. The results of numerical experiments confirm the theoretical findings.
