global existence, regularity and uniqueness of the latter (family) of equation(s) leading us to theorem 2.3. Then in Section 3 we provide further analysis of the family of eikonal equations which lead us to local representations of the solution. In Section 4 we construct first a weak approximation of the solution (in $L^p$ sense), and then extend this to a recursive construction of an $H^{s,p}$-approximation. In Section 5 we provide error estimates by using geometric information. Section 6 points out how the method may be applied for accurate approximation of diffusions, and we finish with a conclusion in Section 7.

Global existence and regularity of the squared Riemannian distance $d^2$
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We shall only sketch the connection between geodesics and the eikonal equation . It is almost standard, and details can be found in [@Ka] and [@Jo]. Our interest here is that the eikonal equation together with careful chosen boundary conditions has a global and unique solution. We shall have two different arguments for uniqueness: one is via uniqueness of an associated diffusion and WKB-representations (or, alternatively, Varadhan’s result, cf. [@V]), but we will have the same insight from an other point of view when we look at local representations 