An adaptive fast multipole accelerated Poisson solver for complex geometries

Askham, Travis; Cerfon, Antoine

doi:10.1016/j.jcp.2017.04.063

Cited by 38 publications

(45 citation statements)

References 44 publications

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“…In this second approach, the complex domain is embedded into a larger, regular domain and the boundary conditions are approximated by a variety of different techniques. Examples include the adaptive fast multipole accelerated Poisson solver (e.g., [4]), which combines boundary and volume integral methods in the larger domain, fictitious domain methods (e.g., [5,6,7,8]) where Lagrange multipliers are applied in order to enforce the boundary conditions, immersed boundary (e.g., [9,10,11,12]), front-tracking (e.g., [13,14,15]) and arbitrary Lagrangian-Eulerian methods (e.g., [16,17,18,19]) utilize separate surface and volume meshes where force distributions are interpolated from the surface to the volume meshes, in a neighborhood of the domain boundary, to approximate the boundary conditions. In addition, a number of specialized methods have been designed to achieve better than first order accuracy in the L ∞ norm.…”

Section: Introductionmentioning

confidence: 99%

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Guo

Lowengrub

2020

Journal of Computational Physics

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The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a larger, regular domain. The original PDE is reformulated using a smoothed characteristic function of the complex domain and source terms are introduced to approximate the boundary conditions. The reformulated equation, which is independent of the dimension and domain geometry, can be solved by standard numerical methods and the same solver can be used for any domain geometry. A challenge is making the method higher-order accurate. For Dirichlet boundary conditions, which we focus on here, current implementations demonstrate a wide range in their accuracy but generally the methods yield at best first order accuracy in , the parameter that characterizes the width of the region over which the characteristic function is smoothed. Typically, ∝ h, the grid size. Here, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in , e.g., O( 2 ) in the L 2 norm and O( p ) with 1.5 ≤ p ≤ 2 in the L ∞ norm. Our analytic results are confirmed numerically in stationary and moving domains where the level set method is used to capture the dynamics of the domain boundary and to construct the smoothed characteristic function.

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Section: Introductionmentioning

confidence: 99%

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Guo

Lowengrub

2020

Journal of Computational Physics

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“…Next, we validate our composite solver for the full stationary problem -the inhomogeneous modified Stokes equations (9). We again use the starfish geometry (127), and let the right hand side F be a simple oscillation,…”

Section: Inhomogeneous Problem Convergencementioning

confidence: 99%

A fast integral equation method for the two-dimensional Navier-Stokes equations

Klinteberg¹,

Askham²,

Kropinski³

2020

Journal of Computational Physics

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The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary conditions are handled naturally, and the ill-conditioning caused by high order terms in the PDE is preconditioned analytically. Despite these advantages, the adoption of integral equation methods has been slow due to a number of difficulties in their implementation. This work describes a complete integral equation-based flow solver that builds on recently developed methods for singular quadrature and the solution of PDEs on complex domains, in combination with several more well-established numerical methods. We apply this solver to flow problems on a number of geometries, both simple and challenging, studying its convergence properties and computational performance. This serves as a demonstration that it is now relatively straightforward to develop a robust, efficient, and flexible Navier-Stokes solver, using integral equation methods.

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“…Methods that have been employed for volume integral evaluation include direct finite element mesh calculations [28,29,30,31], and approximate conversion to a boundary integral using either dual reciprocity [32,33] or interior line integration [34,35]. The embedded boundary method in [36] and the fast Poisson solvers [37,38] exploit an easily constructed regular grid covering the domain, in conjunction with a Fast Multipole Method.…”

Section: Introductionmentioning

confidence: 99%

Boundary integral analysis for non-homogeneous, incompressible Stokes flows

Gray

Jakowski²,

Moore

et al. 2019

Adv Comput Math

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A regular-grid volume-integration algorithm is developed for the non-homogeneous 3D Stokes equation. Based upon the observation that the Stokeslet U is the Laplacian of a function H, the volume integral is reformulated as a simple boundary integral, plus a remainder domain integral. The modified source term in this remainder integral is everywhere zero on the boundary and can therefore be continuously extended as zero to a regular grid covering the domain. The volume integral can then be evaluated on the grid. Applying this method to the Navier-Stokes equations will require obtaining velocity gradients, and thus an efficient algorithm for post-processing these derivatives is also discussed. To validate the numerical implementation, test results employing a linear element Galerkin approximation are presented.

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An adaptive fast multipole accelerated Poisson solver for complex geometries

Cited by 38 publications

References 44 publications

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

A fast integral equation method for the two-dimensional Navier-Stokes equations

Boundary integral analysis for non-homogeneous, incompressible Stokes flows

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