Spectral analysis and spectral symbol of d-variate Qp Lagrangian FEM stiffness matrices

We study the spectral properties of the stiffness matrices coming from the Qp Lagrangian FEM approximation of d-dimensional second order elliptic differential problems; here, p = (p1, . . . , pd) ∈ Nd and pj represents the polynomial approximation degree in the j-th direction. After presenting a construction of these matrices, we investigate the conditioning (behavior of the extremal eigenvalues and singular values) and the asymptotic spectral distribution in the Weyl sense, and we find out the so-called (spectral) symbol describing the asymptotic spectrum. We also study the properties of the symbol, which turns out to be a d-variate function taking values in the space of D(p)×D(p) Hermitian matrices, where D(p) = ∏d j=1 pj . Unlike the stiffness matrices coming from the p -degree B-spline IgA approximation of the same differential problems, where a unique d-variate real-valued function describes all the spectrum, here the spectrum is described by D(p) different functions, that is the D(p) eigenvalues of the symbol, which are well-separated, far away, and exponentially diverging with respect to p and d. This very involved picture provides a clean explanation of: a) the difficulties encountered in designing robust solvers, with convergence speed independent of the matrix size, of the approximation parameters p, and of the dimensionality d; b) the possible convergence deterioration of known iterative methods, already for moderate p and d.