Adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives

An anisotropic error estimator involving only first order derivatives is proposed for the Laplace problem and continuous, piecewise linear finite elements. Upper and lower bounds are presented, the involved constants being independent of the mesh aspect ratio provided the error gradient is equidistributed in the directions of maximum and minimum stretching. An anisotropic adaptive algorithm is then proposed, with aim to equidistribute the error gradient in the directions of maximum and minimum stretching. Numerical results in two and three space dimensions show that the effectivity index is aspect ratio independent on such adapted meshes.