A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are $\mathcal{L}\mathcal{L}^*$ methods. In this work, we argue that all high-order methods in this class should be expected to deliver substandard uniform h-refinement convergence rates. In fact, one may not even see rates proportional to the polynomial order $p > 1$ when the exact solution is a constant function. We show that the convergence rate is limited by the regularity of an extraneous Lagrange multiplier variable which naturally appears via a saddle-point analysis. In turn, limited convergence rates appear because the regularity of this Lagrange multiplier is determined, in part, by the geometry of the domain. Numerical experiments support our conclusions.
翻译:近年来提出了若干非标准限定要素方法,其中每一种方法都来自受PDE限制的规范最小化问题,最显著的例子为$\mathcal{L ⁇ mathcal{L ⁇ {L ⁇ }$的方法。在这项工作中,我们争辩说,这一类中所有高级方法都有望达到低于标准的统一纤维化趋同率。事实上,当精确的解决方案是一个不变的函数时,人们甚至可能看不到与多级顺序$p > 1美元成比例的汇率。我们表明,由于一个超异的拉格朗乘数变量的规律性而使趋同率受到限制,而这种规律性自然地通过挂载点分析而出现。反过来,由于这一拉格朗乘数的规律性部分是由域的几何性决定的,因此出现了有限的趋同率。数字实验支持我们的结论。