Computer Assisted Proof of Spurious Eigensolution for Annular and Eccentric Membranes

Spurious eigensolutions in the boundary integral equation (BIE) or boundary e1ement method (BEM) for doub1yconnected domain problems, the eccentric and annular membranes, are studied analytically and numerically in this paper. For the mathematica1 ana1ysis, we employ the null-field integral equation, the degenerate kernels and the Fourier series to prove the existence of spurious eigensolutions in the continuous system. Examples of eccentric case, annular membrane and general shape are solved by using the null-field equation approach and BEM, respectively. Based on the numerical experiment, computer-assisted proof for the existence of spurious eigenvalues in companion with the trivial outer boundary data is given. The spurious eigenvalue is found to be the true eigenvalue of a circular membrane with the inner radius. The SVD structure for the four influence matrices is examined Also, the trivial outer boundary densities are found in case of spurious eigenva1ues and are shown in the bar chart.