Occurrence Of Spurious Eigenvalues Research Articles

True and spurious eigensolutions for the problems with the mixed-type boundary conditions using BEMs

Boundary integral equations and boundary element methods were employed semi-analytically and numerically to study the occurrence of spurious eigenvalues for the interior Helmholtz equations with the mixed-type boundary conditions. The degenerate kernels of fundamental function and Fourier series were utilized in the null-field integral equation to derive the true and spurious eigenfunctions semi-analytically. The complex-valued boundary element method (BEM), the real-part BEM, the imaginary-part BEM and the multiple reciprocity method were utilized to solve the eigenproblem. The results are compared with those of FEM. A simple case of one-dimensional eigenproblem was also addressed. Moreover, the SVD updating techniques in conjunction with the Fredholm's alternative theorem were adopted to sort out the spurious eigenvalues and extract the true ones. It is emphasized that the occurrence of spurious eigenvalues depends on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary conditions for the problem is. The illustrative examples were verified successfully and the numerical results matched well with those of FEM.

Hydrodynamic stability, the Chebyshev tau method and spurious eigenvalues

The Chebyshev tau method is examined; a numerical technique which in recent years has been successfully applied to many hydrodynamic stability problems. The orthogonality of Chebyshev functions is used to rewrite the differential equations as a generalized eigenvalue problem. Although a very efficient technique, the occurrence of spurious eigenvalues, which are not always easy to identify, may lead one to believe that a system is unstable when it is not. Thus, the elimination of spurious eigenvalues is of great importance. Boundary conditions are included as rows in the matrices of the generalized eigenvalue problem and these have been observed to be one cause of spurious eigenvalues. Removing boundary condition rows can be difficult. This problem is addressed here, in application to the Benard convection problem, and to the Orr-Sommerfeld equation which describes parallel flow. The procedure given here can be applied to a wide range of hydrodynamic stability problems.