论文标题
自适应算法的普通收敛而无需利用可靠性和效率
Plain convergence of adaptive algorithms without exploiting reliability and efficiency
论文作者
论文摘要
我们考虑在有限元方法（FEM）和边界元素方法（BEM）的上下文中的H-自适应算法。在构建块上的相当普遍的假设下，我们对这种算法的求解，估计，标记和完善，在自适应算法将基本的后验误差估计器驱动到零的意义上，我们证明了平淡的收敛性。与文献中可用的结果不同，我们的分析避免了使用任何可靠性和效率估算的使用，但仅依赖于估计量的结构特性，即稳定性，即未精制元素并减少精制元素。因此，新框架因此涵盖了涉及非本地运算符等非本地运算符的问题，例如分数laplacian或边界积分方程，而（当前）（当前）不可用。
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.
