# 1st Leibniz MMS Days - Abstract

**Si, Hang**

*Anisotropic Finite Element Mesh Adaptation via Higher Dimensional Embedding*

In this talk we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding to obtain an anisotropic curvature adapted mesh that fits a complex surface in $mathbbR^3$. In the context of adaptive finite element simulation, the solution (which is an unknown function $f : Omega subset mathbbR^d to mathbbR$) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, $Phi_f(bf x) := (x_1,, ldots, x_d,, s, f(x_1,, ldots, x_d), s, nabla f(x_1,, ldots, x_d))^,t$, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function $f$ itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function $f$. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted $Phi_f(bf x)$ for adaptive finite element analysis. We will provide a serie of experimental tests for piecewise linear interpolation of known functions as well as adaptive finite element solutions of partial differential equations.