Computing Eigenmodes of Elliptic Operators Using Radial Basis Functions

K e y w o r d s Radial basis functions, Eigenvalues, Numerical methods, Laplacian, corner singularities. 1. I N T R O D U C T I O N Many positive properties of radial basis function (RBF) methods have been identified in connection with boundary-value problems (BVPs) [1-4]. They are grid-free numerical schemes very suitable for problems in irregular geometries. They can exploit accurate and smooth represental~ions of the boundary, are very easy to implement, and can be spectrally accurate [5,6]. It would ]be expected tha t the benefits experienced by the use of RBFs in BVPs would carry over for eigenvalue problems. In this article, we formulate and apply an RBF-based method to compute eigenmodes of elliptic operators. Given a linear elliptic second-order partial differential operator L and a bounded region ~ in :~n with boundary 0~, we seek eigenpairs (Au) E (C, C(gt)) satisfying Lu + Au = O, in ~, and L s u = 0, on O~t, (1) *Supported by NSF Grant DMS-0104229. 0898-1221/04/$ see front matter ~) 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2003.08.007 Typeset by A2~S-TEX 562 R.B. PLATTE AND T. A. DRISCOLL where L s is a linear boundary operator of the form L B u = a u + b(n. Vu). (2 ) Here, a and b are given constants and n is the unit outward normal vector defined on the boundary. We assume that gt is open and that the eigenvalue problem is well-posed. We use an interpolating RBF approximation of an eigenfunction of (1) and replace the eigenvalue problem above with a finite-dimensional eigenvalue problem. In order to compute the eigenpairs of the modified system, we approximate the operator L by a matrix that incorporates the boundary conditions and then use standard techniques to find the eigenvalues and eigenvectors of this matrix. As pointed out in [2,7], straightforward RBF approximations are relatively inaccurate near the boundaries, and special attention should be given to this issue. We study the boundary clustering of nodes and the collocation of the PDE on the boundary and verify that they are effective techniques for preventing degradation of the solution near the edges of the domain. We consider the effect of corner singularities in 2-D regions. It is known that some eigenfunctions are not smooth at corners with interior angle 7r/~ where ~ is not integer. As a result, a straightforward RBF approximation is severely degraded in this case. Several authors have exploited explicit information about the singular behavior of an eigenfunction at a corner to propose efficient algorithms to compute eigenvalue for these regions (see, for example, [8-10]). With these efforts as motivation, we append terms to the RBF expansion that approximate the singular behavior of an eigenfunction. To our knowledge, this type of singular augmentation has not been used before with RBFs. The remainder of this article is organized as follows. In the next section, we introduce RBF approximations. In Section 3, we formulate the problem in matrix form. In Section 4, we present the boundary treatment techniques. In Section 5, a scheme for regions with corner singularities is introduced. We carry out several experiments in 2-D regions for the Laplacian operator for a disk, an L-shaped region, and a rhombus as pictured in Figure 1, and compare to finite elements and pseudospectral methods.