A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

Abstract

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some approriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.

Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

“…Hence, the relation (25) makes it possible to extend the P-Arnoldi to this situation. Note that the extraction of Ritz pairs is performed in the standard way, i.e., based on (24). When performing experiments with the resulting P-Arnoldi method for A −1 , we encountered numerical difficulties for larger d. These difficulties are due to the fact that the norms of τ j (H −1 ) grow quickly if H −1 has eigenvalues too far away from the interval [−1, 1].…”

“…We first address the 3D Laplace eigenvalue problem with Dirichlet boundary conditions [24] −∆u = λ 2 u in D ⊂ R 3 , u = 0 on B := ∂D, which can be reformulated by means of the representation formula for the Helmholtz operator as the boundary integral equation…”

Is the Burton–Miller formulation really free of fictitious eigenfrequencies?

“…Therefore, it is not an easy task to solve acoustic eigenproblems using BEM. In order to solve such problems, a number of transform methods have been proposed [11][12][13][14][15][16][17][18][19][20][21], including the dual reciprocity method [11,12], the particular integral method [13,14], the multiple reciprocity method [15] and their applications [16][17][18][19][20]. Also, a numerical eigenvalue analysis by the Galerkin BEM has been carried out in the framework of the concept of eigenproblems for holomorphic Fredholm operatorvalued functions [22].…”

Coupled FE–BE method for eigenvalue analysis of elastic structures submerged in an infinite fluid domain

“…The volume integrals caused by the non-homogeneous term can be transformed into boundary integrals with the help of various methods, such as the dual reciprocity method [23], the particular integral method [24], the multiple reciprocity method [25], and the radial integration method [26]. Further developments and applications of these volume-or domain-integral transform methods can be found, e.g., in References [27][28][29][30][31][32] for acoustic eigenvalue problems and in Reference [33] for fluid-structure interaction eigenvalue problems.Besides the methods mentioned earlier, contour integral methods [34][35][36][37][38] have been recently developed. By virtue of the contour integral methods presented in References [35][36][37][38], a nonlinear eigenvalue problem can be easily converted into a linear one whose dimension is much smaller than the original one.…”