Anisotropic mesh adaptation based upon a posteriori error estimates

An anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is considered. The adaptation method generates anisotropic adaptive meshes as quasiuniform ones in some metric space. The associated metric tensor is computed by means of a posteriori hierarchical error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. The exact solution of the corresponding global error problem can be very costly. However, numerical results show that a few iterations of the symmetric Gauß-Seidel method are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.