@Article{IJNAM-22-860,
author = {Ren , Yiming and Zhao , Shan},
title = {A Multigrid-Based Fourth Order Finite Difference Method for Elliptic Interface Problems with Variable Coefficients},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2025},
volume = {22},
number = {6},
pages = {860--896},
abstract = {<p style="text-align: justify;">The paper introduces a fourth-order augmented matched interface and boundary
(AMIB) method for solving elliptic interface problems with complex interfaces and piecewise
smooth coefficients in two and three dimensions. To resolve the challenge posed by non-constant
coefficients within the AMIB framework, the fast Fourier transform (FFT) Poisson solver of the
existing AMIB methods is replaced by a geometric multigrid method to efficiently invert the
Laplacian discretization matrix. In this work, a fourth order multigrid method will be employed in
the framework of the AMIB method for elliptic interface problems with variable coefficients in two
and three dimensions. Based on a Cartesian mesh, the standard fourth-order finite differences are
employed to approximate the first and second derivatives involved in the Laplacian with variable
coefficients. Near the interface, a fourth-order ray-casting matched interface and boundary (MIB)
scheme is generalized to variable coefficient problems to enforce interface jump conditions in the
corrected finite difference discretization. The augmented formulation of the AMIB allows us to
decouple the interface treatments from the inversion of the Laplacian discretization matrix, so
that one essentially solves an elliptic subproblem without interfaces. A fourth order geometric
multigrid method is introduced to solve this subproblem with a Dirichlet boundary condition,
where fourth order one-sided finite difference approximations are considered near the boundary in
all grid levels. The proposed multigrid method significantly enhances the computational efficiency
in solving variable coefficient problems, while achieving a fourth-order accuracy in accommodating
complex interfaces and discontinuous solutions.</p>},
issn = {2617-8710},
doi = {https://doi.org/10.4208/ijnam2025-1038},
url = {http://global-sci.org/intro/article_detail/ijnam/24298.html}
}