Optimally Convergent Isoparametric \(P^2\) Mesh Generation

This paper describes a framework for the generation of \(P^2\) isoparametric meshes in two dimensions. It is an extension of existing metric-based methods for high-order straight-sided mesh adaptation to curved meshes. Starting with an interpolation error estimate for curved trajectories, we compute an optimal metric field using a generalization of the log-simplex algorithm for high-order metrics. A straight quasi-unit mesh is generated in a frontal approach, then the edges are curved to minimize their length in the chosen metric. Convergence studies are performed on simple test cases. In particular, optimal convergence rates (third order) are reached even when the edges curvature decreases at the same rate as the edge length.