Convergence and Optimality of Adaptive Least Squares Finite Element Methods
Convergence and Optimality of Adaptive Least Squares Finite Element Methods
复制标题
自适应最小二乘有限元方法的收敛性和最优性
DOI：
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发表时间：
2015
影响因子：
2.9
通讯作者：
Eun‐Jae Park
中科院分区：
文献类型：
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作者：
C. Carstensen;Eun‐Jae Park
一阶除最小二乘有限元方法 (LSFEM) 允许通过最小二乘泛函的可计算残差进行直接后验误差控制。本文建立了基于一些等效细化指标的自适应细化策略。由于一阶 div LSFEM 测量 $H$(div) 中的通量误差，因此数据分辨率误差测量右侧 $f$ 的 $L^2$ 范数减去分段多项式近似值 $Pi f$（不带网格尺寸因子）。因此，数据分辨率项既不是振荡也不是高阶的，因此需要特殊的处理，例如通过 Binev 和 DeVore 的阈值第二算法。由此产生的具有单独标记的新型自适应 LSFEM 相对于非线性近似类的概念以最佳速率收敛。
The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in $H$(div), the data resolution error measures the $L^2$ norm of the right-hand side $f$ minus the piecewise polynomial approximation $Pi f$ without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class.