SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1281-1306, June 2022.
Adaptive finite element methods (AFEMs) are an increasingly common means of automatically controlling error in numerical simulations. Proofs of convergence and rate of convergence exist for AFEMs; however, these proofs typically rely upon a nested structure for the sequence of meshes. A metric adaptive finite element method (MAFEM) utilizes the continuous mesh model and instead seeks to optimize a Riemannian metric field for a given cost, from which a mesh is generated. This meshing process results in a sequence of nonnested meshes. In this paper we introduce a proof of convergence for a class of MAFEM, utilizing an optimization statement to relate the error on the sequence of meshes. In addition, we prove that such a sequence of meshes will demonstrate the optimal asymptotic rate of convergence for a given polynomial order. Finally some numerical results demonstrate the performance of the algorithm for a singularly perturbed linear advection diffusion problem.

中文翻译：

---

通过度量优化收敛各向异性网格自适应

SIAM 数值分析杂志，第 60 卷，第 3 期，第 1281-1306 页，2022 年 6 月。
自适应有限元方法 (AFEM) 是一种在数值模拟中自动控制误差的越来越普遍的方法。AFEM 存在收敛证明和收敛速度；然而，这些证明通常依赖于网格序列的嵌套结构。度量自适应有限元方法 (MAFEM) 利用连续网格模型，而是寻求针对给定成本优化黎曼度量场，从中生成网格。此网格划分过程会产生一系列非嵌套网格。在本文中，我们介绍了一类 MAFEM 的收敛证明，利用优化语句来关联网格序列上的误差。此外，我们证明这样的网格序列将证明给定多项式阶的最佳渐近收敛速度。