@Article{NMTMA-5-19,
author = {Armando Coco and Giovanni Russo},
title = {Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2012},
volume = {5},
number = {1},
pages = {19--42},
abstract = {<p style="text-align: justify;">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 (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 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 [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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2011.m12si02},
url = {http://global-sci.org/intro/article_detail/nmtma/5926.html}
}