Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method
Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method
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DOI：
10.1016/j.jcp.2015.07.036
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发表时间：
2015-10
期刊：
影响因子：
--
通讯作者：
Lunji Song;Zhimin Zhang
中科院分区：
文献类型：
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作者：
Lunji Song;Zhimin Zhang
针对一类拟线性椭圆型问题的过罚对称内罚不连续Galerkin解，提出了一种多项式保持恢复方法。作为一种后处理方法，多项式保持恢复对于规则型和v型网格下的线性和二次元，以及满足条件（ε, σ）的一般网格都是超收敛的。利用求平均技术，证明了求平均解的多项式保持恢复方法是超收敛的，并满足与拟合有限元方法相似的估计。我们直接从不连续解中推导出恢复梯度的超收敛性，并自然构造了一个后验误差估计量。因此，基于恢复梯度的后验误差估计是渐近精确的。给出了与我们的分析相一致的大量数值结果。
A polynomial preserving recovery method is introduced for over-penalized symmetric interior penalty discontinuous Galerkin solutions to a quasi-linear elliptic problem. As a post-processing method, the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns, as well as general meshes satisfying Condition (ϵ, σ). By means of the averaging technique, we prove the polynomial preserving recovery method for averaged solutions is superconvergent, satisfying similar estimates as those for conforming finite element methods. We deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact. Extensive numerical results consistent with our analysis are presented.