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.

翻译：权重是允许推广拉格朗日有限元的几何自由度。它们定义在特定支撑区域上的积分，可通过微分形式和积分理论得到充分理解，并属于有限元外微积分框架。本文利用该形式体系，以识别对有限元近似具有吸引力的支撑区域为目标。为此，我们研究了相关参数矩阵序列（矩阵阶数随网格尺寸趋零而趋于无穷）。利用标准Toeplitz机制和广义局部Toeplitz（GLT）理论，我们描述了条件数和谱全局行为，从而确定了权重的最优选择。此外，基于离散化参数并与共轭梯度法结合，我们提出并测试了专用预处理器。考虑的模型问题是一维拉普拉斯算子（包括常系数和变系数情形）。通过数值可视化与实验测试的严格讨论，我们论证了权重诱导基相较于标准拉格朗日基的优势。结论部分列出了尚未解决的问题和未来研究方向，特别关注高维情形。