Convergent finite element approximations of surface evolution with relaxed minimal deformation

The finite element approximation of surface evolution under an external velocity field is studied. An artificial tangential motion is designed by using harmonic map heat flow from the initial surface onto the evolving surface. This makes the evolving surface have minimal deformation (up to certain relaxation) from the initial surface and therefore improves the mesh quality upon discretization. By exploiting and utilizing an intrinsic cancellation structure in this formulation and the role played by the relaxation term, convergence of the proposed method in approximating surface evolution in the three-dimensional space is proved for finite elements of degree \ (k 4\). One advantage of the proposed method is that it allows us to prove convergence of numerical approximations by using the normal vector of the computed surface in the numerical scheme, instead of evolution equations of normal vector (as in the literature). Another advantage of the proposed method is that it leads to better mesh quality in some typical examples, and therefore prevents mesh distortion and breakdown of computation. Numerical examples are presented to illustrate the convergence of the proposed method and its advantages in improving the mesh quality of the computed surfaces.