The numerical linear algebra of weights. Part I: from the spectral analysis to conditioning and preconditioning in the one-dimensional Laplacian case

Weights are geometrical degrees of freedom that allow to generalize Lagrangian finite elements. They are defined through integrals over specific supports, well understood in terms of differential forms and integration, and lie within the framework of finite element exterior calculus. We adopt this formalism with the target of identifying supports that are appealing for a finite element approximation, describing weights in terms of a single parameter in the third- and fourth-order methods. To do so, we study the related parametric matrix-sequences, with the matrix order tending to infinity as the mesh size tends to zero. Using the generalized locally Toeplitz theory, we analyze the performance of weights-based finite elements on an elliptic operator. In particular, for degrees 3 and 4, we identify an optimal value for the weights location, which sits in a rather large interval where weights give rise to better conditioned stiffness matrices. With this at hand, we propose and test ad hoc preconditioners, in dependence of the discretization parameters and in connection with conjugate gradient method. The model problem we consider is a one-dimensional Laplacian, both with constant and non-constant coefficients. Numerical visualizations and experimental tests are reported and critically discussed, showing a confidence interval for the choice of the parameter.