35K10 Second-order parabolic equations
We study the approximation of Wasserstein gradient structures
by their finite-dimensional analog. We show that simple
finite-volume discretizations of the linear Fokker-Planck
equation exhibit the recently established entropic gradient-flow
structure for reversible Markov chains. Then we reprove the
convergence of the discrete scheme in the limit of vanishing
mesh size using only the involved gradient-flow structures.
In particular, we make no use of the linearity of the equations
nor of the fact that the Fokker-Planck equation is of second order.
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.