  In this paper we present a one dimensional second order accurate method to
solve Elliptic equations with discontinuous coefficients on an arbitrary
interface. Second order accuracy for the first derivative is obtained as well.
The method is based on the Ghost Fluid Method, making use of ghost points on
which the value is defined by suitable interface conditions. The multi-domain
formulation is adopted, where the problem is split in two sub-problems and
interface conditions will be enforced to close the problem. Interface
conditions are relaxed together with the internal equations, leading to an
iterative method on all the set of grid values (inside points and ghost
points). A multigrid approach with a suitable definition of the restriction
operator is provided. The restriction of the defect is performed separately for
both sub-problems, providing a convergence factor close to the one measured in
the case of smooth coefficient and independent on the magnitude of the jump in
the coefficient. Numerical tests will confirm the second order accuracy.
Although the method is proposed in one dimension, the extension in higher
dimension is currently underway.
