How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms
- Kampen, Jörg
2010 Mathematics Subject Classification
- 65N99 35F20 35K10
- length of geodesic, regular polynomial interpolation
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 $ds^2=g_ijdx^idx^j$ 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 approximations of solutions of linear parabolic partial differential equations leading to obvious applications in finance, physics and other sciences.