Spectral collocation solutions to multiparameter Mathieu’s system

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

“…3.13]. However, since discretization with the ChC requires much smaller matrices than with finite differences (see, e.g., [16,Ex. 5]), N is usually small enough so that the ill-conditioning is not an issue.…”

“…The problem along with the appropriate boundary conditions is treated as a 2EP in [16] and solved numerically very accurately and efficiently with the Chebyshev collocation. As discussed in the introduction, we will extend the results in [16] considerably in several ways. methods from this paper are suitable for all problems of the form (11), regardless if they come from separation of variables or not.…”

“…Compared with [16], this paper presents advances in several directions. First, we consider several other very important problems.…”

“…Yet, it was not until [16] that this approach was actually used to compute the eigenfrequencies of an elliptic membrane. In [16] it is shown that the two-parameter approach has certain advantages with respect to the accuracy as well as to the required computational time and can be used in practice to numerically evaluate a large number of eigenfrequencies.…”

“…Second, while a Jacobi-Davidson style solver [19,20] and standard implicitly restarted Arnoldi [37] were used in [16] to solve the 2EPs, we speed up the computations in this paper even more by exploiting a new fast Sylvester-Arnoldi algorithm for 2EPs from [29]; this algorithm is briefly described in Section 6. Third, we compare with, and improve on several available results in the literature in the numerical examples in Section 7.…”