[1] 
U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf, and R. Rusu.
A finite element method for surface restoration with smooth boundary
conditions.
Computer Aided Geometric Design, 21(5):427445, 2004. [ bib  .pdf 1 ] In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C^{1}continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundary conditions to ensure continuity of the normal field. The corresponding L^{2}gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as system of two coupled second order equations. As it is wellknown, fourth order problems require an implicit time discretization. Here a semiimplicit approach is presented which allows large timesteps. For the discretization of the boundary condition, two different numerical methods are introduced. Finally, we show applications to different surface restoration problems.
