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SIAM Journal on Numerical Analysis
Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension
SIAM Journal on Scientific Computing
Efficient management of parallelism in object-oriented numerical software libraries
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A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries
SIAM Journal on Scientific Computing
A Fast Iterative Algorithm for Elliptic Interface Problems
SIAM Journal on Numerical Analysis
The immersed interface method using a finite element formulation
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A Cartesian grid embedded boundary method for Poisson's equation on irregular domains
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A boundary condition capturing method for Poisson's equation on irregular domains
Journal of Computational Physics
The immersed interface method for the Navier-Stokes equations with singular forces
Journal of Computational Physics
A second-order-accurate symmetric discretization of the Poisson equation on irregular domains
Journal of Computational Physics
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SIAM Journal on Scientific Computing
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SIAM Journal on Scientific Computing
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A numerical method for solving variable coefficient elliptic equation with interfaces
Journal of Computational Physics
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An immersed interface method for simulating the interaction of a fluid with moving boundaries
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Journal of Computational Physics
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Journal of Computational Physics
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Journal of Computational Physics
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Journal of Computational Physics
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Journal of Computational Physics
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Journal of Computational Physics
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Journal of Computational Physics
SIAM Journal on Numerical Analysis
Journal of Computational Physics
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Journal of Computational Physics
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Journal of Computational Physics
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Journal of Computational Physics
A parallel multigrid Poisson solver for fluids simulation on large grids
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SIAM Journal on Numerical Analysis
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Hi-index | 31.46 |
We present a numerical method for the variable coefficient Poisson equation in three-dimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric positive definite and equal to the standard 7-point finite difference Poisson stencil away from embedded interfaces and boundaries. Numerical evidence suggests second order accuracy in the L^~-norm. Our approach improves the treatment of Dirichlet and jump constraints in the recent work of Bedrossian et al. [1] and introduces innovations necessary in three dimensions. Specifically, we construct new constraint-based Lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations; we provide a method for cell-local polyhedral approximation to the zero isocontour surface of a level set needed for three-dimensional embedding; and we show that the new Lagrange multiplier spaces naturally lead to a class of easy-to-implement multigrid methods that achieve near optimal efficiency, as shown by numerical examples. For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints. As in [1], this is used in a discontinuity removal technique that yields the standard 7-point stencil across the interface and only requires a modification to the right-hand side of the linear system.