JournalsifbVol. 2 , No. 4DOI 10.4171/ifb/24

Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs

  • Gerhard Dziuk

    Universität Freiburg, Germany
  • Klaus Deckelnick

    Otto-von-Guericke-Universität Magdeburg, Germany
Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs cover

Abstract

The efficient numerical simulation of the curvature-driven motion of interfaces is an important tool in several free- boundary problems. We treat the case of an interface which is given as a graph. The highly non-linear problem is discretized in space by piecewise linear finite elements. Although the problem is not in divergence form it can be written in a variational form which allows the use of the modern adaptive techniques of finite elements. The time discretization is carried out in a semi-implicit way such that in every time step a linear system with symmetric positive matrix has to be solved. Optimal error estimates are proved for the fully discrete problem under the assumption that the time-step size is bounded by the spatial grid size.