An RBF-FD closest point method for solving PDEs on surfaces

Petras, Argyrios; Ling, Leevan; Ruuth, Steven J.

doi:10.1016/j.jcp.2018.05.022

Cited by 53 publications

(48 citation statements)

References 42 publications

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“…are expressed as in [35], augmented with polynomial terms as described in [43]. Specifically, the matrix A and the vector B have the form…”

Section: Description Of the Methodsmentioning

confidence: 99%

“…Using the method of lines approach and applying an implicit time stepping method in (12) leads to an over-determined system. For example, using the techniques described in [35], the application of the backward differentiation formula BDF2 yields…”

Section: Description Of the Methodsmentioning

confidence: 99%

“…If this condition is satisfied, then the application of step 3 for z k is necessary. For more details on the RBF-FD calculation, we refer to Section 3.1 and [35]. Additional information on the resampling step of the GBPM can be found in [37].…”

Section: Pde Solutionmentioning

confidence: 99%

“…Recently, a closest point method using finite differences derived from radial basis functions (RBF-FDs) has been proposed [35]. This method, namely RBF-CPM, uses a smaller computational tube than the original CPM and is particularly flexible with respect to the choice of points used to form the finite difference stencils.…”

Section: Introductionmentioning

confidence: 99%

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A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

Petras

Ling

Piret

et al. 2019

Journal of Computational Physics

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The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys.228 (8): 2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.

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“…are expressed as in [35], augmented with polynomial terms as described in [43]. Specifically, the matrix A and the vector B have the form…”

Section: Description Of the Methodsmentioning

confidence: 99%

Section: Description Of the Methodsmentioning

confidence: 99%

Section: Pde Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

Petras

Ling

Piret

et al. 2019

Journal of Computational Physics

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“…For example, see [2] for the closed-form formulations of orthogonal projection method. In addition, several recent articles presented hybrid methods that incorporate features of RBF approach alongside embedding methods [53].…”

Section: History Of Rbf Approaches To Solving Pdesmentioning

confidence: 99%

Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions

Delengov¹

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This dissertation has been duly read, reviewed, and critiqued by the Committee listed below, which hereby approves the manuscript of Vladimir Delengov as fulfilling the scope and quality requirements for meriting the degree of Doctor of Philosophy.In this work, numerical approaches based on meshless methods are proposed to obtain eigenmodes of elliptic operators on manifolds, and their performance is compared against existing alternative methods. Radial Basis Function (RBF)based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly's formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces. Prospective extensions of the methods include application to problems with boundary conditions and incorporating a multi-layer approach in order to improve accuracy. The latter is justified by an asymptotic expansion of eigenvalues of Laplace operator on a thin ring and a thin shell. This work is dedicated to my beloved fiancée Sara, who always supported me unconditionally and believed in me no matter what. v

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Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

Jiang

Harlim

2021

Comm Pure Appl Math

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In this paper, we extend the class of kernel methods, the so‐called diffusion maps (DM) and its local kernel variants to approximate second‐order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the ghost point diffusion maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points on the boundary. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values at the ghost points. This adjustment is analogous to the classical ghost point method in a finite‐difference scheme for solving PDEs on flat domains. As opposed to the classical DM, which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary. Applying the consistent GPDM estimator to solve well‐posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We numerically validate the proposed mesh‐free PDE solver on various problems defined on simple submanifolds embedded in Euclidean spaces as well as on an unknown manifold. Numerically, we also found that the GPDM is more accurate compared to DM in solving elliptic eigenvalue problems on bounded smooth manifolds. © 2021 Wiley Periodicals LLC.

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An RBF-FD closest point method for solving PDEs on surfaces

Cited by 53 publications

References 42 publications

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions

Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

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