Error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity |
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Authors: | Xiaobing Feng Michael Neilan Andreas Prohl |
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Institution: | 1.Department of Mathematics,The University of Tennessee,Knoxville,USA;2.Mathematisches Institut,Universit?t Tübingen,Tübingen,Germany |
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Abstract: | This paper proposes 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 (J
Diff Geom 59:353–437), 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 involves a small parameter ɛ; a rigorous analysis is presented to study well-posedness and convergence of the scheme under certain mesh-constraints, and
optimal rates of convergence are verified. We then prove 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 ɛ both tend to zero. Computational results are provided to show the efficiency of the proposed finite element method and to
numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow. Numerical studies
are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter ɛ for optimal convergence of the proposed scheme. Finally, a numerical convergence study for another approach recently proposed
by R. Moser (The inverse mean curvature flow and p-harmonic functions. preprint U Bath, 2005) for approximating the inverse mean curvature flow via p-harmonic functions is also included. |
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Keywords: | 65N30 65N12 35J70 83C57 |
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