SIAM Journal on Scientific and Statistical Computing
A level set approach for computing solutions to incompressible two-phase flow
Journal of Computational Physics
Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L2-norm
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
Computation of three dimensional dendrites with finite elements
Journal of Computational Physics
Introduction to “A numerical method for solving incompressible viscous flow problems”
Journal of Computational Physics - Special issue: commenoration of the 30th anniversary
Preconditioned multigrid methods for unsteady incompressible flows
Journal of Computational Physics
Journal of the ACM (JACM)
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
Journal of Computational Physics
A multigrid tutorial: second edition
A multigrid tutorial: second edition
A remark on computing distance functions
Journal of Computational Physics
Robust multigrid methods for nonsmooth coefficient elliptic linear systems
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. III: linear algebra
Multigrid
Journal of Computational Physics
A second-order-accurate symmetric discretization of the Poisson equation on irregular domains
Journal of Computational Physics
A Boundary Condition--Capturing Multigrid Approach to Irregular Boundary Problems
SIAM Journal on Scientific Computing
A node-centered local refinement algorithm for Poisson's equation in complex geometries
Journal of Computational Physics
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
Journal of Computational Physics
Journal of Scientific Computing
A sharp interface finite volume method for elliptic equations on Cartesian grids
Journal of Computational Physics
Level-set, penalization and cartesian meshes: A paradigm for inverse problems and optimal design
Journal of Computational Physics
An efficient fluid-solid coupling algorithm for single-phase flows
Journal of Computational Physics
Journal of Scientific Computing
Journal of Computational Physics
A second order virtual node method for elliptic problems with interfaces and irregular domains
Journal of Computational Physics
Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces
Journal of Computational Physics
Journal of Computational Physics
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In this paper we present a numerical method for solving elliptic equations in an arbitrary domain (described by a level-set function) with general boundary conditions (Dirichlet, Neumann, Robin, etc.) on Cartesian grids, using finite difference discretization and non-eliminated ghost values. A system of N"i+N"g equations in N"i+N"g unknowns is obtained by finite difference discretization on the N"i internal grid points, and second order interpolation to define the conditions for the N"g ghost values. The resulting large sparse linear system is then solved by a multigrid technique. The novelty of the papers can be summarized as follows: general strategy to discretize the boundary condition to second order both in the solution and its gradient; a relaxation of inner equations and boundary conditions by a fictitious time method, inspired by the stability conditions related to the associated time dependent problem (with a convergence proof for the first order scheme); an effective geometric multigrid, which maintains the structure of the discrete system at all grid levels. It is shown that by increasing the relaxation step of the equations associated to the boundary conditions, a convergence factor close to the optimal one is obtained. Several numerical tests, including variable coefficients, anisotropic elliptic equations, and domains with kinks, show the robustness, efficiency and accuracy of the approach.