Journal Article10.1137/S1064827501389849
The Immersed Interface/Multigrid Methods for Interface Problems
Loyce Adams,Zhilin Li +1 more
131
TL;DR: The second order maximum principle preserving finite difference scheme for linear parabolic equations, using the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives is developed.
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Abstract: New multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic PDEs that involve interfaces and discontinuities. For elliptic interface problems, the multigrid solver developed in this paper works while some other multigrid solvers do not. For linear parabolic equations, we have developed the second order maximum principle preserving finite difference scheme in this paper. We use the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives. Numerical examples are also presented.
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Citations
High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources
TL;DR: This paper introduces a novel high order interface scheme, the matched interface and boundary (MIB) method, for solving elliptic equations with discontinuous coefficients and singular sources on Cartesian grids by appropriate use of auxiliary line and/or fictitious points.
400
Immersed-Interface Finite-Element Methods for Elliptic Interface Problems with Nonhomogeneous Jump Conditions
TL;DR: A class of new finite- element methods, called immersed-interface finite-element methods, is developed to solve elliptic interface problems with nonhomogeneous jump conditions to provide fast simulation of interface dynamics that does not require remeshing.
280
A new multiscale finite element method for high-contrast elliptic interface problems
TL;DR: A new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces is introduced.
249
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
Sining Yu,Y. C. Zhou,Guo-Wei Wei +2 more
TL;DR: This work generalizes the matched interface and boundary method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems, and introduces primary and secondary fictitious values to deal with sharp-edged interfaces.
205
An overview of the immersed interface method and its applications
TL;DR: The immersed interface method for various problems is introduced, its recent advances and related software packages are discussed, and some of its applications are reviewed.
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Randall J. LeVeque,Zhilin Li +1 more
TL;DR: In this paper, the authors developed finite difference methods for elliptic equations of the form \[
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William L. Briggs,Van Emden Henson,Steve F. McCormick +2 more
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The immersed interface method for the Navier-Stokes equations with singular forces
Zhilin Li,Ming-Chih Lai +1 more
TL;DR: An immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain is proposed based on a second-order projection method with modifications only at grid points near or on the interface.
428
Black Box Multigrid
TL;DR: The end result is code, BOXMG, in which one need only specify the (logically rectangular, positive definite) matrix problem; BOXMG does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution.
367
Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients
Zhilin Li,Kazufumi Ito +1 more
TL;DR: New finite difference methods using Cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients, singular sources, and nonsmooth or even discontinuous solutions to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques.