A posteriori error estimates and adaptive finite elements for meshes with high aspect ratio: application to elliptic and parabolic problems

Following the framework of Formaggia and Perotto (Numer.

Math. 2001 and 2003), anisotropic a posteriori error estimates have been

proposed for various elliptic and parabolic problems. The error in the

energy norm is bounded above by an error indicator involving the matrix

of the error gradient, the constant being independent of the mesh aspect

ratio. The matrix of the error gradient is approached using

Zienkiewicz-Zhu error estimator. Numerical experiments show that the

error indicator is sharp. An adaptive finite element algorithm which

aims at producing successive triangulations with high aspect ratio is

proposed. Numerical results will be presented on various problems such

as diffusion-convection, Stokes problem, dendritic growth.