How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms

2008/11/11 by Joerg Kampen, Kampen, Joerg
Mathematics · #35F20 #65N99 #Differential Equations and Boundary Problems #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.0811.1734

openalex publication_date 2008/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element ds²=g_ijdxⁱdx^j of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximation of solutions of parabolic partial differential equations leading obvious applications to finance and physics.

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