Convergence of meshless Petrov-Galerkin method using radial basis functions
Introduction
In the last decade, there has been some advance developments in applying the radial basis functions (RBFs) for the numerical solutions of various types of partial differential equations (PDEs). The initial development was due to the pioneering work of Kansa [1] who directly collocated the RBFs for the approximated solutions of the equations. In general, the Kansas method has several advantage over the widely used FEM, in that
- (1)
It is a truly meshless in which the collocation points can be choose freely (no connectivity between points is required as FEM), Hence the complicated meshing problem has been avoid.
- (2)
It is spatial dimension independent which can easily be extended to solve high dimensional problems.
Despite the many special attractive features of RBFs, it is known that most of the RBFs are globally defined basis functions. This means that the resulting matrix for interpolation is dense and can be highly ill-conditioned, especially for a large number of interpolation points in 3D. This poses serious stability problems and high computational cost. At the same time, the CS-RBFs also have several difficulties: (i) the accuracy and efficiency depend on the scale of the support and determining the scale of support is uncertain; (ii) the convergence rate of CS-RBFs is low. In order to obtain a sparse matrix system, the support needs to be small; then the interpolation error become unacceptable. When the support is large enough to make the error acceptable, the matrix system becomes dense and the advantages to the traditional RBFs are lost.
In the society of the meshless RBFs methods, there are many techniques to circumvent these problems. To my knowledge, the most important techniques might be DDM [19], [4], [13], [26], Precondition methods [17], MLS [8], [9], [10], [11], [12]. Certainly more useful techniques also can be found in [21], [18], [5], [16], [24], [2], [3], [27].
The most popular methods in solving PDEs using RBFs might be the collocation method and the Galerkin method [22]. The collocation method is simple but not stable and difficult to analyze [14]. At the same time, the computation of the Galerkin method is too complex to be used in the practice. In this paper, we will provide the meshless Petrov-Galerkin method, in which the trial space is generated by global supported RBFs, the test space is generated by compactly supported RBFs. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method.
The organization of this paper is as follows: in Section 2, we introduce the Petrov-Galerkin method, the convergence of the meshless Petrov-Galerkin method will be stated in Section 3, some techniques to deal with the boundary conditions would be provided in Section 4, and the numerical examples are provided in Section 5, and the Section 6 with conclusion and remarks.
Section snippets
Petrov-Galerkin method
For a bounded domain Ω with C1-boundary ∂Ω we consider problems of the formwhere aij, c ∈ L∞(Ω), i, j = 1 , … , n, f ∈ L2(Ω), aij, h ∈ L∞(∂Ω), i, j = 1 , … , n, g ∈ L2(∂Ω) and ν denotes the unit normal vector to the boundary ∂Ω. The matrix A(x) = (aij(x)) is assumed to be uniformly elliptic on Ω, i.e. there is a constant γ such that for all x ∈ Ω and all α ∈ RdWe further require that c ⩾ 0 and h ⩾ 0, and that at
Convergence of the meshless Petrov-Galerkin method using radial basis functions
In this section, we will prove that, under certain assumptions, the result obtained using this meshless Petrov-Galerkin method combined with radial basis functions converge to the solution of the PDEs gradually.
Firstly, we introduce a few useful definitions as below: Definition 1 Let H be a Hilbert space, u ∈ H be a nonzero vector, H1 ⊂ H be a subspace of H (it might not be completed), the projection of u on subspace H1 is defined as Remark 1 It is well known that for all x ∈ H (H is a Hilbert
Dealing with the boundary conditions
For the sake of simplicity, we consider the following PDEs:where Ω is a bounded domain with C1-boundary.
If we have the Robin boundary conditions:we can use the Petrov-Galerkin method directly as stated in Section 2. If we have the Dirichlet boundary conditionsthen we have two methods to deal with it:
- (1)
We can use the Petrov-Galerkin method at the inner points, and use the collocation method at the boundary points simply.
- (2)
We can approximate the Dirichlet boundary value
Numerical experiments
In this section, we will solve the following partial differential equations:using our meshless Petrov-Galerkin method, where Ω = [a, b] × [c, d], f = −2sin(x + y), g = sin(x + y). Obviously, the weak formulation of PDEs (13) iswhereIn this numerical example, we choose n = nx × ny collocation points all together (nx points on each horizontal line and ny points on each vertical line), and
Conclusion and remarks
In the literature of FEM, the Galerkin method to solve partial differential equations might be the most popular one. But in the society of meshless RBFs, there are much more dominant methods, such as FSM [1], BKM [7], DRM [20] MLS [8] and collocation method [15], one reason is that the Galerkin method is not so efficient as in FEM because more integrals in the whole domain should be calculated. Our meshless Petrov-Galerkin method might be a remedy. It would be more efficient if we use the
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