A DRBEM approximation of the Steklov eigenvalue problem

In this study, we propose a novel approach based on the dual reciprocity boundary element method (DRBEM) to approximate the solutions of various Steklov eigenvalue problems. The method consists in weighting the governing differential equation with the fundamental solutions of the Laplace equation where the definition of interior nodes is not necessary for the solution on the boundary. DRBEM constitutes a promising tool to characterize such problems due to the fact that the boundary conditions on part or all of the boundary of the given flow domain depend on the spectral parameter. The matrices resulting from the discretization are partitioned in a novel way to relate the eigenfunction with its flux on the boundary where the spectral parameter resides. The discretization is carried out with the use of constant boundary elements resulting in a generalized eigenvalue problem of moderate size that can be solved at a smaller expense compared to full domain discretization alternatives. We systematically investigate the convergence of the method by several experiments including cases with selfadjoint and non-selfadjoint operators. We present numerical results which demonstrate that the proposed approach is able to efficiently approximate the solutions of various mixed Steklov eigenvalue problems defined on arbitrary domains.


Manguoğlu, Murat (Society for Industrial & Applied Mathematics (SIAM), 2019-01-01)
We propose a two-level nested preconditioned iterative scheme for solving sparse linear systems of equations in which the coefficient matrix is symmetric and indefinite with a relatively small number of negative eigenvalues. The proposed scheme consists of an outer minimum residual (MINRES) iteration, preconditioned by an inner conjugate gradient (CG) iteration in which CG can be further preconditioned. The robustness of the proposed scheme is illustrated by solving indefinite linear systems that arise in t...
Fundamental solution for coupled magnetohydrodynamic flow equations
Bozkaya, Canan; Tezer, Münevver (Elsevier BV, 2007-06-01)
In this paper, a fundamental solution for the coupled convection-diffusion type equations is derived. The boundary element method (BEM) application then, is established with this fundamental solution, for solving the coupled equations of steady magnetohydrodynamic (MHD) duct flow in the presence of an external oblique magnetic field. Thus, it is possible to solve MHD duct flow problems with the most general form of wall conductivities and for large values of Hartmann number. The results for velocity and ind...
A model for the computation of quantum billiards with arbitrary shapes
Erhan, Inci M.; Taşeli, Hasan (Elsevier BV, 2006-10-01)
An expansion method for the stationary Schrodinger equation of a three-dimensional quantum billiard system whose boundary is defined by an arbitrary analytic function is introduced. The method is based on a coordinate transformation and an expansion in spherical harmonics. The effectiveness is verified and confirmed by a numerical example, which is a billiard system depending on a parameter.
A coupled numerical scheme of dual reciprocity BEM with DQM for the transient elastodynamic problems
Bozkaya, Canan (Wiley, 2008-11-12)
The two-dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first- and the second-order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second...
A class of orthogonal polynomials suggested by a trigonometric Hamiltonian : Symmetric states
Taşeli, Hasan (Springer Science and Business Media LLC, 2004-05-01)
A new subclass of the Jacobi polynomials arising in the exact analytical solution of the one-dimensional Schrodinger equation with a trigonometric potential has been introduced. The polynomials which consist of a free parameter are not ultraspherical polynomials and have been simply named the T - polynomials since they are generated by a trigonometric Hamiltonian. In certain sense, it is shown that the T - polynomials can be regarded as a generalisation of the airfoil polynomials or the Chebyshev polynomial...
Citation Formats
Ö. Türk, “A DRBEM approximation of the Steklov eigenvalue problem,” ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS, pp. 232–241, 2021, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/70201.