We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by a fixed number of few smoothing steps. Even though these intermediate solutions are far from the exact algebraic solutions, their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost. We quantify rigorously how the error propagates throughout the algorithm, and we provide a connection with classical a posteriori error analysis. A series of numerical experiments highlights the efficiency and the computational speedup of S-AFEM.

中文翻译：

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基于平滑自适应有限元法的准最优网格序列构建

我们针对线性、二阶、椭圆偏微分方程 (PDE) 提出了一种基于平滑迭代 (S-AFEM) 的自适应有限元方法 (AFEM) 新算法。该算法的灵感来自 V 循环多重网格方法的上升阶段：我们使用延长步骤替换经典 AFEM 中间循环中的准确代数解，然后是固定数量的几个平滑步骤。尽管这些中间解与精确的代数解相距甚远，但它们的后验误差估计会产生一种细化模式，该模式与经典 AFEM 生成的模式基本等效，但计算成本却相当可观。我们严格量化错误如何在整个算法中传播，我们提供了与经典后验误差分析的联系。一系列数值实验突出了 S-AFEM 的效率和计算速度。