Is it possible to transform continuity boundary condition to boundary value problem in magnetostatic?

The goal is to find the flux density distribution at all points on a 2d plane, (a magnetostatic problem).
the problem consists of simple geometry areas(the interfaces are lines across only one dimension) with different permeabilities and some have excitation (current and magnetization).

As we know this requires the Laplace and Poisson equation and all boundary conditions to be satisfied. the boundary conditions in magnetostatic requires that the parallel component of field intensity and normal component of flux density on the boundary between two adjacent media be continuous. however this leads to complicated equations.

But if the values on the boundaries were known, the solution is much more simpler; i.e.: is it possible to somehow find the potential on the boundaries first before solving the whole problem? If it is, what is the equation to find the value on only the boundaries?
Does this question regard method of imaging or boundary element method ? However I have no idea about these methods. If there is a helpful resource, I would be glad to know about. thanks.