Spectral analysis and spectral symbol of d-variate mathbb Q-oldsymbol p Lagrangian FEM stiffness matrices

We study the spectral properties of the stiffness matrices coming from the approximation of a d-dimensional second order elliptic differential problem by the Qp Lagrangian finite element method (FEM); here, p = (p1, . . . , pd) and pj represents the polynomial approximation degree in the jth direction. After presenting a construction of these matrices, we investigate the conditioning and the spectral distribution in the Weyl sense, and we determine 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 N(p) × N(p) Hermitian matrices, where N(p) = p1*...*pd. Unlike the stiffness matrices coming from the p -degree B-spline isogeometric analysis approximation of the same differential problem, where a unique d-variate real-valued function describes all the spectrum, here the spectrum is described by N(p) different functions, i.e., the N(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 an 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 convergence deterioration of known iterative methods, already observed in practice for moderate p and d.