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Weights are geometrical degrees of freedom that allow to generaliseLagrangian finite elements. They are defined through integrals over specificsupports, well understood in terms of differential forms and integration, andlie within the framework of finite element exterior calculus. In this work weexploit this formalism with the target of identifying supports that areappealing for finite element approximation. To do so, we study the relatedparametric matrix-sequences, with the matrix order tending to infinity as themesh size tends to zero. We describe the conditioning and the spectral globalbehavior in terms of the standard Toeplitz machinery and GLT theory, leading tothe identification of the optimal choices for weights. Moreover, we propose andtest ad hoc preconditioners, in dependence of the discretization parameters andin connection with conjugate gradient method. The model problem we consider isa onedimensional Laplacian, both with constant and non constant coefficients.Numerical visualizations and experimental tests are reported and criticallydiscussed, demonstrating the advantages of weights-induced bases over standardLagrangian ones. Open problems and future steps are listed in the conclusivesection, especially regarding the multidimensional case.

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