A priori anisotropic mesh adaptation driven by a higher dimensional embedding

We generalize the higher embedding approach proposed in Lévy and Bonneel (2013) to generate an adapted mesh matching the intrinsic directionalities of an assigned function. In more detail, the original embedding map between the physical (lower dimensional) and the embedded (higher dimensional) setting is modified to include information associated with the function and with its gradient. Then, we set an adaptive procedure, driven by the embedded metric but performed in the lower dimensional setting, which results into an anisotropic adapted mesh of the physical domain. The effectiveness of the proposed procedure is extensively investigated on several two-dimensional test cases, involving both analytical functions and finite element approximations of differential problems. The preliminary verification in three dimensions corroborates the robustness of the method.