Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems

Abstract

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 suit- able 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 prob- lem. Interface conditions are relaxed together with the internal equations (following the approach proposed in(10) in the case of smooth coefficients), leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid ap- proach 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 (12) and it will be carried out by combining the discretization of(10) with the multigrid approach of(11) for Elliptic problems with non-eliminated boundary conditions in arbitrary domain. AMS subject classifications: 35J25, 65N06, 65N55