The least-squares finite element method (LSFEM) has drawn much attention with desirable properties, such as always symmetric positive-definite stiffness matrix and approximation of primal and non-primal variables with equivalent accuracy. Despite a condition number comparable to that of Galerkin FEM, of $O\left(h^{-2}\right)$, it is sometimes found that LSFEM with low-order elements does not show a satisfactory accuracy of solution. In this report, we propose an easy-to-implement technique that optimizes the condition number of the stiffness matrix. The least-squares functional is applied to the first-order governing equations and a linear combination of the weak form domain and boundary equations, with normalization (i.e., weighting) parameters is constructed. The impact of these parameters on the condition number of the resulting global stiffness matrix is proved to be independent of mesh size or shape function order. By use of a global optimization scheme that tunes these parameters, we observe notable decrease in the condition number compared to cases where the least-squares principle is directly applied to the governing equations, and hence convergence is significantly improved. It is further shown that the technique can be applied to the elemental stiffness matrix for a computationally efficient means of determining normalization parameters. To show the general applicability of the method, we also apply this technique to solve for a two-dimensional (2D) lid-driven flow problem and to calculate the electromagnetic field distribution in a 2D rectangular resonant cavity. In both problems, the LSFEM solution accuracy is poor prior to optimization of weighting parameters, and accuracy does not increase much by mesh refinement alone. Moreover, the process can be generalized to Galerkin FEM and other methods to improve convergence properties.

最小二乘有限元法 (LSFEM) 因其理想的特性而备受关注，例如始终对称的正定刚度矩阵以及具有等效精度的原始变量和非原始变量的近似值。尽管条件数与 Galerkin FEM 相当，但$哦\left(H^{-2}\right)$，有时会发现具有低阶单元的 LSFEM 没有显示出令人满意的求解精度。在本报告中，我们提出了一种易于实施的技术，可以优化刚度矩阵的条件数。最小二乘函数应用于一阶控制方程和线性组合使用归一化（即加权）参数构造弱形式域和边界方程。这些参数对所得全局刚度矩阵的条件数的影响被证明与网格尺寸或形状函数阶数无关。通过使用调整这些参数的全局优化方案，与将最小二乘原理直接应用于控制方程的情况相比，我们观察到条件数显着减少，因此收敛性得到显着改善。进一步表明，该技术可以应用于元素刚度矩阵，用于确定归一化参数的计算有效手段。为了显示该方法的普遍适用性，二维矩形谐振腔中的电磁场分布。在这两个问题中，LSFEM 求解精度在优化权重参数之前较差，并且仅通过网格细化并不能提高精度。此外，该过程可以推广到 Galerkin FEM 和其他方法以提高收敛性。