Abstract
We present a numerical method for the computation of the \(n\)-dimensional Ricci–DeTurck flow. The Ricci flow is a partial differential equation (PDE) deforming a time-dependent metric on a closed Riemannian manifold in proportion to its Ricci curvature. The Ricci–DeTurck flow is a reparametrization of this flow using the harmonic map flow in order to get a strictly parabolic PDE. Our numerical method is based on the assumption that the manifold is embeddable into \(\mathbb R^{n+1}\) as a differentiable manifold. By this means, it is possible to do computations in the Euclidean coordinates of the ambient space. A weak formulation of the Ricci–DeTurck flow is derived such that it only contains tangential gradients. A spatial discretization of this formulation with finite elements on polyhedral hypersurfaces and a semi-implicit time discretization lead to an algorithm for computing the Ricci–DeTurck flow. We have performed numerical tests for two- and three-dimensional hypersurfaces using piecewise linear finite elements. The generalization to non-orientable hypersurfaces of higher codimensions is still open.
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Acknowledgments
This work was supported by the German Research Foundation DFG via the SFB/TR 71 “Geometric Partial Differential Equations”. We thank Gerhard Dziuk for helpful discussions.
