Variational grid adaptation based on the minimization of local truncation error: time-independent problems

• Authors: Giovanni Lapenta
• Venue: Journal of Computational Physics
• Year: 2004

A new approach to grid adaptation is presented. The method is based on two established foundations. First, the method is based upon variational grid adaptation, retaining all the well-known properties of robustness and regularity. Second, the adaptation method presented here is based on a general definition of the error detector obtained from the moving finite element (MFE) method. The error detector is general, applicable to any given problem, and does not require any a priori knowledge of the solution or of the physical behaviour of the system under investigation. The primary theoretical contribution of the present work is in establishing a link between various adaptation methods previously regarded as different and unrelated. We show that they all derive from the same approach and are all equivalent in the sense that the same grid is generated by all of them for the same problem, once the monitor functions are chosen according to our approach. The primary practical contribution of the present work is in prescribing a rigorous monitor function for previously published adaptation strategies. The choice proposed here is shown to outperform previous heuristic choices. The method is tested in a series of elliptic problems, where the adaptation strategy presented here can improve the accuracy by orders of magnitude.