Convergence analysis of the adaptive finite element method with the red-green refinement

In this paper, we analyze the convergence of the adaptive conforming P-1 element method with the red-green refinement. Since the mesh after re. ning is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the initial mesh, we propose a modified red-green refinement and prove the convergence of the associated adaptive method under a much weaker condition on the initial mesh (Condition B).