Generalized Locally Toeplitz matrix-sequences and approximated PDEs on submanifolds: the flat case

In the present paper, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions where the operator is the Laplace-Beltrami operator Δ over (Formula presented.), Ω being an open and bounded submanifold of (Formula presented.), (Formula presented.). We will take into consideration the classical (Formula presented.) Finite Elements, in the case of Friedrichs-Keller triangulations, leading to sequences of matrices of increasing size. We are interested in carrying out a spectral analysis of the resulting matrix-sequences. The tools for our derivations are mainly taken from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz (GLT) matrix-sequences. The current contribution is only quite an initial step, where a general programme is provided, with partial answers leading to further open questions: indeed the analysis is performed on special flat submanifolds and hence there is room for wide generalizations, with a final picture which is still unclear with respect to, e.g. the role of the submanifold curvature.