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：权是允许推广拉格朗日有限元的几何自由度。它们通过特定支撑上的积分定义，在微分形式与积分的框架下得到充分理解，并隶属于有限元外微分的体系。本文利用这一形式化方法，旨在识别对有限元逼近具有吸引力的支撑。为此，我们研究了相关的参数矩阵序列，其中矩阵阶数随网格尺寸趋近于零而趋于无穷大。我们基于标准Toeplitz工具与GLT理论描述了条件化与谱全局行为，从而确定了权的最优选择。此外，我们提出并测试了依赖于离散化参数并与共轭梯度法相结合的自适应预条件算子。所考虑的模型问题是一维拉普拉斯算子（包括常系数与非常系数情形）。通过数值可视化与实验测试的呈现与批判性讨论，展示了基于权的基相较于标准拉格朗日基的优势。在结论部分，我们列出了开放性问题与未来步骤，特别是针对多维情形。