论文标题
高阶LL*（FOSLL*）有限元方法的先验错误分析
A priori error analysis of high-order LL* (FOSLL*) finite element methods
论文作者
论文摘要
近年来已经提出了许多非标准有限元方法，每种方法都来自特定类别的PDE限制的规范最小化问题。最著名的示例是$ \ Mathcal {l} \ Mathcal {l}^*$方法。在这项工作中，我们认为应该期望该类别中的所有高阶方法提供不合标准的H-重新收敛率。实际上，当精确解决方案是恒定函数时，甚至可能不会看到与多项式订单$ p> 1 $成比例的速率。我们表明，收敛速率受到额外的Lagrange乘数变量的规律性的限制，该变量自然会通过鞍点分析出现。反过来，出现有限的收敛速率，因为该拉格朗日乘数的规律性部分通过域的几何形状确定。数值实验支持我们的结论。
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.
