Error reduction of the adaptive conforming and nonconforming finite element methods with red–green refinement

In this paper, we analyze the convergence of the adaptive conforming and nonconforming $P_1$ finite element methods with red–green refinement based on standard Dörfler marking strategy. Since the mesh after refining is not nested into the one before, the usual Galerkin-orthogonality or quasi-orthogonality for newest vertex bisection does not hold for this case. To overcome such a difficulty, we develop some new quasi-orthogonality instead under certain condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive methods by establishing the reduction of some total errors. To weaken the condition on the initial mesh, we propose a modified red–green refinement and prove the convergence of the associated adaptive methods under a much weaker condition on the initial mesh (Condition B). Furthermore, we also develop an initial mesh generator which guarantee that all the interior triangles are equilateral triangles (satisfy Condition A) and the other triangles containing at least one vertex on the boundary satisfy Condition B.