Vectơ trọng số cho hàm poisson

Đặng Thị Oanh Tạp chí KHOA HỌC & CÔNG NGHỆ 78(02): 63 - 66

63

RBF STENCILS FOR POISSON EQUATION

Dang Thi Oanh*

Faculty of Information Technology - TNU

ABSTRACT

In the paper we present a method for finding the weight vector called stencil with the help of RBF

interpolation. This stencil is the foundation for constructing meshless finite difference scheme for

boundary value problems. The results of numerical experiments show that the numerical solution

obtained by RBF-FD with the stencils generated by Gauss RBF interpolation is much more

accurate then the solution obtained by FEM.

Keyword: Radial Basis Function (RBF), meshless method, shape parameter, stencil

INTRODUCTION*

Because of the difficulties to create, maintain

and update complex meshes needed for the

standard finite difference, finite element or

finite volume discretisations of the partial

differential equations, meshless methods

attract growing attention. In particular, strong

form methods such as collocation or

generalised finite differences are attractive

because they avoid costly numerical

integration of the non-polynomial shape

functions on non-standard domains often

encountered in those meshless methods that

are based on the weak formulation of PDE.

Thanks to their excellent local approximation

power, radial basis functions are an ideal tool

to produce numerical differentiation stencils

for the Laplacian and other partial differential

operators on irregular centres, without any

need for a mesh. This leads to exceptionally

promising RBF based generalised finite

difference method.

The essense of the method is to find a weight

vector called stencil corresponding a row of

stiffness matrix in FEM. In this paper we give

formulas for finding the stencil based on

RBF, and then we use them for discretising

the Poisson equation on a nonuniform set of

centres. In [1], [2] for the above purpose we

used RBF interpolation with additional

polynomial term and performed many

numerical examples in complicated geometry

domains with several RBF. In this paper we

*

Email: [email protected]

concentrate on RBF interpolation without

additional polynomial term and test the

Poisson equation on the disk domain with

RBF-Gaussian. The results of numerical

experiments show that the numerical solution

obtained by RBF-FD with the stencil

generated by Gauss RBF interpolation is

much more accurate then the solution

obtained by FEM.

The paper is organised as follows. In Section

2 we describe stencils from RBF

interpolation. Section 3 is devoted RBF-FD

discretisation of Poisson equation and finally

Section 4 we provide the results of the

numerical tests with the near-optimal shape

parameter on the disk domain.

STENCILS FROM RBF INTERPOLATION

Definition 1 (stencil) Let D be a linear

differential operator, and

 

n

i i X x

1

 a fixed

irregular set of centres in Rd

. A linear

numerical differentiation formula for the

operator D,





n

i

i i Du x w x u x

1

( ) ( ) ( ), (1)

is determined by the weights

wi

=

w (x)

i

.

The vector

 

T

w w wn

, ,

 1 

is called stencil.

Definition 2 (positive definite function) A

continuous function

:

d   R R is called

positive semi-definite if, for all

n N

, all

sets of pairwise distinct centers

 

d X  x1

,, xn  R

, and all c = (c1, c2,

…, cn)  R

n

, the quadratic form

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