A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions

Crockett, Robert; Colella, Phillip; Graves, Daniel

doi:10.1016/j.jcp.2010.12.017

Cited by 60 publications

(62 citation statements)

References 25 publications

(31 reference statements)

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“…the solution itself is given at the boundary. The case of imposing Neumann or Robin or jump boundary conditions is not the focus of this paper and we refer the interested reader to [32,63,11,146,75,77,104,78,114,55,50,150,83] and the references therein. Finally, typical scientific applications exhibit solutions with different length scales.…”

Section: Introductionmentioning

confidence: 99%

“…[110,81,80] and the references therein. We note that block structured AMR solvers, aided by efficient multigrid solvers (see [32] and the references therein), have advantages in that the entire grid structure may be stored efficiently, which may speed up the execution time. However, they do not have the flexibility of Octrees and require more grid points and therefore computational time.…”

Section: Introductionmentioning

confidence: 99%

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High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

2012

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We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain's boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain's boundary and (iii) a quadtree/octree nodebased adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

2012

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“…Note that the surface is tracked by a level-set without particles in this experiment, and that the blue color indicates fluid cells, rather than depicting the actual fluid domain. separated fluid component, similar to [44], [45], [46], [47]. Conceptually, we build a distinct coarse grid for each separated fluid component, with the coarse grids being allowed to overlap.…”

Section: Multigrid Solvermentioning

confidence: 99%

Large-Scale Liquid Simulation on Adaptive Hexahedral Grids

Ferstl

Westermann

Dick

2014

IEEE Trans. Visual. Comput. Graphics

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Abstract-Regular grids are attractive for numerical fluid simulations because they give rise to efficient computational kernels. However, for simulating high resolution effects in complicated domains they are only of limited suitability due to memory constraints. In this paper we present a method for liquid simulation on an adaptive octree grid using a hexahedral finite element discretization, which reduces memory requirements by coarsening the elements in the interior of the liquid body. To impose free surface boundary conditions with second order accuracy, we incorporate a particular class of Nitsche methods enforcing the Dirichlet boundary conditions for the pressure in a variational sense. We then show how to construct a multigrid hierarchy from the adaptive octree grid, so that a time efficient geometric multigrid solver can be used. To improve solver convergence, we propose a special treatment of liquid boundaries via composite finite elements at coarser scales. We demonstrate the effectiveness of our method for liquid simulations that would require hundreds of millions of simulation elements in a non-adaptive regime.

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“…However, the one-sided quadratic interpolations used to compute the fluxes along the boundary yield an asymmetric system. See [82,83] for a more recent 3-dimensional version applied to Poisson's equation and the heat equation. In [84], Oevermann and Klein proposed a second order finite volume method for interface problems, and simplified and extended their method to 3-dimensions in [85].…”

Section: Introductionmentioning

confidence: 99%

“…However, less general multigrid algorithms specially tuned to the particular discretization method may outperform a black-box multigrid solver; see, for example, [28,86]. Some methods lend themselves to using relatively straightforward extensions of standard geometric multigrid techniques, including both mortar finite element methods [7,9] and embedded methods [48,[81][82][83], usually with special attention being paid near irregular features. Many of the works describing IIM-based discretizations [26][27][28]22] utilize a multigrid solver with a grid hierarchy defined geometrically but incorporate algebraic techniques in the remaining components (coarse-grid operators and grid transfer operators).…”

Section: Introductionmentioning

confidence: 99%

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

Hellrung

Wang

Sifakis

et al. 2012

Journal of Computational Physics

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Keywords: Elliptic interface problems Embedded interface methods Virtual node methods Variational methods Multigrid methods a b s t r a c tWe present a numerical method for the variable coefficient Poisson equation in threedimensional 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 1 -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.

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A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions

Cited by 60 publications

References 25 publications

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

Large-Scale Liquid Simulation on Adaptive Hexahedral Grids

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

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