About theoretical and practical impact of mesh adaptation on approximation of functions and PDE solutions

We try to formalize and study how mesh adaptation improves the approximation of interpolated functions or of PDE solutions. We first define an adaptive solution, in the sense that the pair (mesh,function) satisfies a non-linear coupled equation. In order to build optimal mesh adaptation strategies, we also define a functional model, the ʹcontinuous metricʹ, which leads to propose the best mesh for a given function and a given norm. We then describe how convergence of adaptive solutions can be better than for non-adaptive ones; this involves some recent refinements concerning what we called early capturing of details, a specific property of good adaptive strategies. We give some typical numerical illustrations. Convergence properties depend very much on how mesh adaptation is performed and we exhibit theoretical limits for the maximum order of accuracy reachable for some family of mesh adaptation methods.