The reconstruction of a vector field by finite difference methods

The problem of reconstruction of a vector field, given in a refracting medium, is considered. The approach to an approximate solution of the problem consisting in combination of variational, geometrical and finite difference methods is implemented. A polynomial approximation of the vector field, obtained by method of least squares, can contain an arbitrary potential part. A determination of the potential part of a required approximation of the field therefore is necessary by a solution of a homogeneous boundary value problem for the equation of Beltrami- Laplace in a bounded domain with an arbitrary configuration of boundaries. The formulated boundary value problem is solved by finite difference methods with usage of a continuous isometry of an original domain with the given Riemannian metric on a domain with the reference configuration of a boundary. For such domain the effective finite difference and iterative methods designed, the necessary evaluations and numerical experiments are produced.