Adaptive Finite Element Method for General Second‐Order Nonlinear Elliptic Problems

ABSTRACT In this paper, we analyze the simplest and most standard adaptive finite element method (AFEM) for general second‐order nonlinear elliptic problems and provide an analysis of its convergence. In this AFEM algorithm, we only mark the elements according to the error estimator and refine them without the interior property. The AFEM is based on a natural and computationally efficient residual‐based a posteriori error estimator for the finite element approximations. The reliability and efficiency of the error estimator are established by deriving global upper bounds and local lower bounds of the approximation error in the ‐norm. With the global upper bound, quasi‐orthogonality, and error indicator reduction property, we show that AFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. Any prescribed error tolerance is thus achieved in a finite number of cycles. Numerical experiments are also provided to illustrate the performance of the adaptive algorithm.