Corrected fundamental solution for numerical solution of elliptic PDEs

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Abstract

Corrected fundamental solution (CFS) is a meshless method for homogeneous elliptic problems that corrects the density function in a simple layer potential integral. In the CFS method, we apply a new expansion of density function with variable coefficients which are approximated in a finite subspace of a complete space. These coefficients are determined by the moving least square method (MLS), using a suitable weight function that its support is in the real and artificial domain.

Introduction

The method of fundamental solution (MFS) is a numerical method for homogeneous elliptic problems that needs knowing and having related fundamental solutions [6], [3], [8]. It is a boundary meshless method. This method originally presented by Kupradze and Aleksidze [6]. First, simple and double layer integrals is approximated on artificial boundary and then by expanding its density function in a finite subspace of a complete basis, one have an approximation solution depend on the related problem. Unknown coefficients of the density function must be found by collocating the approximate solution on the boundary. By selecting some boundary points equal to the coefficients, one can find the coefficients of the density function using the collocation method on the real boundary. Instead of the collocation method, many boundary points may be chosen, more than the unknown density coefficients and so, by using the least square method the unknown density coefficients can be found.

In 1968, Shepard [9] presented the moving least square method (MLS) in a simple and low order form entitled Shepard Interpolant. Then in 1981, Lancaster and Salkauskas [7] generalized the MLS method and extended it to higher order [4].

In this paper, we change the density function expansion or series, and correcting it similar to the MLS approximation together with the variable coefficients. These expansions need a finite subspace of a complete basis such as polynomial and trigonometric functions.

The rest of this paper is structured as follows: Section 2 introduces the MFS. Section 3 is a review on the MLS method and in Section 4, we explained our method, the CFS. In Section 5, we presented a 2–D numerical example. Section 6 introduces a typical adaptivity. Section 7 gives our concluding remarks.

Section snippets

Method of fundamental solution

Fundamental solution of a problem is potential of a point source charge. In the MFS, one knows FS previously. This method is a boundary meshless method and needs an artificial domain, which is greater than real domain.

This method is as follows:

Let ΩRd, d = 1, 2 or 3 be an open domain and ∂Ω = Γ be its boundary.

Consider the following homogeneous elliptic boundary value problem:Lu(x)=0,xΩ,u(x)=g(x),xΓ.

Fundamental solution of the model problem (1) for a fixed point y  Ω is the solution ofLH(x,y)=δ(x,y

Moving least square method (MLS)

Let u:ΩR, where ΩRd,d=1,2or3 be an unknown continuous function that we try to approximate it by having some data point. Given xj  Ω, j = 1, 2,  , n, an irregular distribution of nodes in the domain and uj = u(xj), j = 1, 2,  , n. Let P(x) be a given m-dimensional base, for example, in 1-D case, let PT(x) = {1, x,  , xm−1}. Define local approximationu˜y(x)=PT(x)a(y),where y  Ω is fixed and the coefficient vector a(y) = [a1(y), a2(y),  , am(y)]T should be found. Let wi(x), i = 1, 2,  , n be a suitable weight. By minimizing

Corrected fundamental solution (CFS)

Referring to the Section 3, for a fixed point y  Ω, let σ(x, y) be a corrected density function emerged of the MLS method and {ϕj(y)}j=1n be a finite subspace of a complete space. Then, the density function can be approximated in the following form:σ(x,y)=j=1ncj(x)ϕj(y),xΩ¯.

The solution of the problem (1) based on the single layer potential by this density function, becomesu˜(x)=Γ^σ(x,y)H(x,y)ds(y),xΩ¯,where Γ^ is the artificial boundary and H(x, y) is the fundamental solution of related

A numerical example

Let Ω¯=[0,1]×[0,1] be closure of the real domain and Ω^¯=[-1.0,2.0]×[-1.0,2.0] be closure of the artificial domain. The real and artificial domain are rectangular, and their boundary comprise of four lines.

Here, our model problem isΔu(x,y)=0,(x,y)Ω,u(0,y)=u(1,y)=u(x,1)=0,u(x,0)=sin(πx),x[0,1],which is the Laplace problem and Dirichlet boundary conditions. The exact solution of this problem is u(x, y) = sin(πx)(cosh(πy)  coth(π)sinh(πy)), where (x,y)Ω¯. The FS of the Laplace operator, isH(x,y,ξ,η)

Adaptivity

We can add another base function like {p}=1ne, such as polynomial base over another base {ψj}j=1n that was introduced in (7), enrichment of the approximation (6) can be done. So, an alternative approximation becomesu˜(x)=j=1ncjψj(x)+=1nedp(x),xΩ¯.

It is important to note that the above approximation is a solution of the model problem (1), ifLp(x)=0,xΩ¯,for=1,2,,ne.

For example, if we want to enrich the bases of the model problem (36), a finite subset of the base {Rz}=0 can be

Concluding remarks

  • Computational task of the CFS is more than the MFS. Because, the inverse of n × n matrix for each evaluation point must be calculated.

  • Both the MFS and the CFS require neither domain nor boundary discretization. They are boundary meshless methods (see [5]).

  • Radial form of the MFS and the CFS is intensive to dimensionality of the problem and thus is very attractive to high-dimensional problems.

  • The correction function in reproducing kernel particle method (RKPM) [1] can be considered for any change

Acknowledgements

The authors would like to thank Prof. C. S. Chen of Department of Mathematical Sciences, University of Nevada Las Vegas, for giving his unpublished preprint “Scientific Computing with Radial Basis Functions” and Dr. Mehdi Dehghan of Amirkabir University of Technology, for his revision and his corrections.

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