We consider $h$-adaptive algorithms in the context of the finite element method and the boundary element method. Under quite general assumptions on the building blocks SOLVE, ESTIMATE, MARK and REFINE of such algorithms we prove plain convergence in the sense that the adaptive algorithm drives the underlying a posteriori error estimator to zero. Unlike available results in the literature, our analysis avoids the use of any reliability and efficiency estimate but relies only on structural properties of the estimator, namely stability on nonrefined elements and reduction on refined elements. In particular, the new framework thus also covers problems involving nonlocal operators like the fractional Laplacian or boundary integral equations, where (discrete) efficiency is (currently) not available.

中文翻译：

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在不利用可靠性和效率的情况下实现自适应算法的简单收敛

我们在有限元法和边界元法的背景下考虑$h$-自适应算法。在对此类算法的构建块 SOLVE、ESTIMATE、MARK 和 REFINE 的非常一般的假设下，我们证明了在自适应算法将基础后验误差估计器驱动为零的意义上的简单收敛。与文献中可用的结果不同，我们的分析避免使用任何可靠性和效率估计，而仅依赖于估计器的结构特性，即非精炼元素的稳定性和精炼元素的减少。特别是，新框架因此还涵盖了涉及非局部算子的问题，例如分数拉普拉斯或边界积分方程，其中（离散）效率（当前）不可用。