A fully optimal anisotropic mesh adaptation method based on a hierarchical error estimator

We present a new two-dimensional mesh adaptation method which produces optimal meshes for all quadratic functions, positive definite or indefinite. The method also leads naturally, and without any specific modification, to anisotropic meshes when the solution allows it. In its simplest form, the method starts with a linear finite element solution and requires the recovery of its gradient which is used to build a new quadratic reinterpolation of the solution. The error is estimated as the difference between the linear solution and its quadratic reinterpolation. Our method does not depend, a priori, on the partial differential equation at hand in contrast to residual methods. It however strongly depends on the quality of the gradient recovery. The method is also easily generalized to higher order finite element solution which is also an improvement over most metric based adaptation methods. This estimated error is used to govern local modifications of the mesh in order to attain a prescribed level of error in L2-norm or a prescribed number of nodes. The H1-seminorm of the estimated error is then minimized in order to get an optimal mesh. We also show that the optimal mesh depends only on the solution itself and very little on the differential operator of the problem at hand. Numerical examples are presented and discussed.