A Note on Eigenvalues and Singular Values of Variable Toeplitz Matrices and Matrix-Sequences, with Application to Variable Two-Step BDF Approximations to Parabolic Equations

The use of variable grid BDF methods for parabolic equations leads to structures that are called variable (coefficient) Toeplitz. Here, we consider a more general class of matrix-sequences and we prove that they belong to the maximal ∗-algebra of generalized locally Toeplitz (GLT) matrix-sequences. Then, we identify the associated GLT symbols and GLT momentary symbols in the general setting and in the specific case, by providing in both cases a spectral and singular value analysis. More specifically, we use the GLT tools in order to study the asymptotic behaviour of the eigenvalues and singular values of the considered BDF matrix-sequences, in connection with the given non-uniform grids. Numerical examples, visualizations, and open problems end the present work.