Finite Element Approximations of the Inverse Mean Curvature Flow Arising from the General Relativity

This paper develops and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in [14], represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature. We first propose a finite element method for a regularized flow which characterized by a small parameter ε, and derive optimal order error estimates with explicit dependence on ε for the finite element solution. We then prove the uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter ε both tends to zero. Numerical experiment results are provided not only to show the efficiency of the proposed finite element method but to numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow as predicted and proved by Huisken and Ilmanen in [14].