We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.

中文翻译：

---

关于自适应最小二乘有限元方法的简单收敛的简短说明

我们表明，在PDE算子，网格细化和标记策略的弱条件下，由规范最小二乘函数驱动的自适应最小二乘有限元方法收敛。与先前的工作相反，我们的普通收敛既不依赖于足够精细的初始网格，也不依赖于对标记参数的严格限制。最后，我们证明了如果使用压缩迭代求解器来获得近似解（例如，具有最佳预处理器的预处理共轭梯度法），则收敛仍然有效。结果适用于涵盖各种模型问题的相当抽象的框架。