This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

Moving interfaces–and in the stationary case, free boundaries–are ubiquitous in our
environment and daily life. They are at the basis of many physical, chemical, and also …

There are several interesting examples of equations governing motion of hypersurfaces
bounding two phases of materials in various sciences. Such a hypersurface is called an …

This review concerns the computation of curvature-dependent interface motion governed by
geometric partial differential equations. The canonical problem of mean curvature flow is that …

Curvature driven surface evolution plays an important role in geometry, applied mathematics
and in the natural sciences. In this paper geometric evolution equations such as mean …

The aim of this paper is to develop a functional-analytic framework for the construction of
level set methods, when applied to shape optimization and shape reconstruction problems …

We consider curves in \mathbbR^n moving by the gradient flow for elastic energy, ie, the L 2
integral of curvature. Long-time existence is proved in the two cases when a multiple of …

We propose a structure-preserving parametric finite element method (SP-PFEM) for
discretizing the surface diffusion of a closed curve in two dimensions (2D) or a surface in …

We present a finite element approximation of motion by minus the Laplacian of curvature
and related flows. The proposed scheme covers both the closed curve case, and the case of …

This paper describes an approach to smooth the surface and improve the quality of
quadrilateral/hexahedral meshes with feature preserved using geometric flow. For …