Hessian-free metric-based mesh adaptation via geometry of interpolation error

The article presents analysis of a new methodology for generating meshes minimizing $L^p$-norms of the interpolation error or its gradient, $p>0$. The key element of the methodology is the construction of a metric from node-based and edge-based values of a given function. For a mesh with $N_h$ triangles, we demonstrate numerically that $L^\infty$-norm of the interpolation error is proportional to $N_h^{-1}$ and $L^\infty$-norm of the gradient of the interpolation error is proportional to $N_h^{-1/2}$. The methodology can be applied to adaptive solution of PDEs provided that edge-based a posteriori error estimates are available.