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We compute the length of geodesics on a Riemannian manifold by regularpolynomial interpolation of the global solution of the eikonal equation relatedto the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithmapproximates the length functional in arbitrarily strong Sobolev norms. Errorestimates are obtained where the geometric information is used. It is pointedout how the algorithm can be used to get accurate approximation of solutions ofparabolic partial differential equations leading obvious applications tofinance and physics.

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