不利用可靠性和效率的自适应算法的简单收敛
Plain convergence of adaptive algorithms without exploiting reliability and efficiency
摘要：
我们认为h-自适应算法的上下文中的有限元法（FEM）和边界元方法（BEM）。在相当一般的假设下，这些算法的积木求解，估计，标记和细化，我们证明了普通的收敛性的意义上说，自适应算法驱动器的基本后验误差估计为零。与文献中的结果不同，我们的分析避免了使用任何可靠性和效率估计，但只依赖于估计的结构特性，即非细化元素的稳定性和细化元素的减少。特别是，新的框架，因此也涵盖了涉及非本地运营商的问题，如分数拉普拉斯算子或边界积分方程，其中（离散）效率（目前）是不可用的。
英文摘要：
We consider h-adaptive algorithms in the context of the finite element method (FEM) and the boundary element method (BEM). 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 only relies on structural properties of the estimator, namely stability on non-refined elements and reduction on refined elements. In particular, the new framework thus covers also problems involving non-local operators like the fractional Laplacian or boundary integral equations, where (discrete) efficiency is (currently) not available.
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