Optimal stencils in Sobolev spaces

Davydov, Oleg; Schaback, Robert

doi:10.1093/imanum/drx076

Cited by 18 publications

(31 citation statements)

References 32 publications

(46 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The convergence order of the RBF-FD schemes is limited by the smoothness of the employed kernel and the geometry of the RBF centers [35]. In this paper, we employ Gaussian RBFs so that the smoothness of the kernel will not be a limiting factor.…”

Section: Parametersmentioning

confidence: 99%

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

Petras

Ling

Ruuth

2018

Journal of Computational Physics

View full text Add to dashboard Cite

Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231 (14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.

show abstract

Section: Parametersmentioning

confidence: 99%

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

Petras

Ling

Ruuth

2018

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…Therefore the · -minimal formulas are scalable in the sense of [14], which is helpful for the computation of w, see Section 6.1.…”

Section: Minimal Formulas and Growth Functionsmentioning

confidence: 99%

“…The best convergence order as h → 0 achieved in these experiments is h 4 for all sets X h i , i = 1, 2, 3, and it is attained by the formulas of polynomial exactness order q = 6 or 7. The fact that the formulas of exactness order 7 do possess better convergence speed than those of order 6 is explained by the finite smoothness of f 1 and H 7 (R 2 ), see [14]. There are significant differences between the sets X h 1 , X h 2 and X h 3 with respect to the question which q is preferable, especially in the pre-asymptotic setting where h is not excessively small.…”

Section: Errors Of Minimal Formulas Of Various Exactness Ordersmentioning

confidence: 99%

Minimal numerical differentiation formulas

Davydov

Schaback

2018

Numer. Math.

Self Cite

View full text Add to dashboard Cite

We investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector. Error bounds are given in terms of a growth function that carries the information about the geometry of the centers. Specific forms of weighted ℓ 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments. The results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods.

show abstract

“…In this section, we describe parameter selection for our method. Given a linear operator L of order θ and an RBF-FD differentiation rule (x j , w j ) n j=1 , we have the following error estimate for RBF-FD based formulas that reproduce a polynomial of degree [10]:…”

Section: Parameter Selectionmentioning

confidence: 99%

RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces

Shankar

Narayan

Kirby

2018

Journal of Computational Physics

View full text Add to dashboard Cite

We present a new method for the solution of PDEs on manifolds M ⊂ R d of co-dimension one using stable scale-free radial basis function (RBF) interpolation. Our method involves augmenting polyharmonic spline (PHS) RBFs with polynomials to generate RBF-finite difference (RBF-FD) formulas. These polynomial basis elements are obtained using the recently-developed least orthogonal interpolation technique (LOI) on each RBF-FD stencil to obtain local restrictions of polynomials in R 3 to stencils on M. The resulting RBF-LOI method uses Cartesian coordinates, does not require any intrinsic coordinate systems or projections of points onto tangent planes, and our tests illustrate robustness to stagnation errors. We show that our method produces high orders of convergence for PDEs on the sphere and torus, and present some applications to reaction-diffusion PDEs motivated by biology.

show abstract

Optimal stencils in Sobolev spaces

Cited by 18 publications

References 32 publications

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

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

Minimal numerical differentiation formulas

RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces

Contact Info

Product

Resources

About