The numerical linear algebra of weights: from the spectral analysis to conditioning and preconditioning in the Laplacian case

Weights are geometrical degrees of freedom that allow to generalise 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. In this work we exploit this formalism with the target of identifying supports that are appealing for finite element approximation. To do so, we study the related parametric matrix-sequences, with the matrix order tending to infinity as the mesh size tends to zero. We describe the conditioning and the spectral global behavior in terms of the standard Toeplitz machinery and GLT theory, leading to the identification of the optimal choices for weights. Moreover, 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 onedimensional Laplacian, both with constant and non constant coefficients. Numerical visualizations and experimental tests are reported and critically discussed, demonstrating the advantages of weights-induced bases over standard Lagrangian ones. Open problems and future steps are listed in the conclusive section, especially regarding the multidimensional case.