New robust nonconforming finite elements of higher order
New robust nonconforming finite elements of higher order
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新的鲁棒高阶非相容有限元
DOI：
10.1016/j.apnum.2011.11.005
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发表时间：
2012
影响因子：
2.8
通讯作者：
P. Zajac
中科院分区：
文献类型：
--
作者：
M. Köster;A. Ouazzi;F. Schieweck;S. Turek;P. Zajac
我们证明了，如果网格序列仍然是形状规则的，但不再由渐近仿射等价的网格单元组成，则现有的四边形高阶非协调有限元的逼近阶数会降低。我们把二阶非协调有限元作为防止这种降阶的一族新的高阶方法的成员来研究。我们提出了一种新的方法，该方法利用非协调胞泡函数来丰富参考单元上的原始多项式空间，最后通过静态凝聚将其去除。在泊松问题的情况下，给出了逼近误差和一致性误差的最优估计，这意味着离散化误差的最优阶。此外，我们讨论了已知的在高阶单元情况下防止降阶的非参数方法，其中基函数定义为原始网格单元上的多项式。对于所得到的线性离散系统的有效处理，我们数值分析了基于正则全阶网格变换算子的相应几何多重网格求解器的收敛。基于几种基准配置，对于标量泊松问题和不可压缩的Navier-Stokes方程(代表这些非协调有限元的理想应用领域)，我们展示了所讨论的新方法的高数值精度、灵活性和效率，并已在FeatFlow软件(www.Featflow.de)中成功实现。结果表明，所提出的有限元-多重网格组合(连同间断压力近似)对于高效的模拟工具来说是非常有利的，特别是对于不可压缩流动问题。
We show that existing quadrilateral nonconforming finite elements of higher order exhibit a reduction in the order of approximation if the sequence of meshes is still shape-regular but consists no longer of asymptotically affine equivalent mesh cells. We study second order nonconforming finite elements as members of a new family of higher order approaches which prevent this order reduction. We present a new approach based on the enrichment of the original polynomial space on the reference element by means of nonconforming cell bubble functions which can be removed at the end by static condensation. Optimal estimates of the approximation and consistency error are shown in the case of a Poisson problem which imply an optimal order of the discretization error. Moreover, we discuss the known nonparametric approach to prevent the order reduction in the case of higher order elements, where the basis functions are defined as polynomials on the original mesh cell. Regarding the efficient treatment of the resulting linear discrete systems, we analyze numerically the convergence of the corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible Navier–Stokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software (www.featflow.de). The presented results show that the proposed FEM-multigrid combinations (together with discontinuous pressure approximations) appear to be very advantageous candidates for efficient simulation tools, particularly for incompressible flow problems.
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DOI：
10.1051/m2an:2002022
发表时间：
2002-05
期刊：
Mathematical Modelling and Numerical Analysis
影响因子：
--
作者：
F. Schieweck
通讯作者：
F. Schieweck
高阶有限元离散化对多重网格收敛的影响
DOI：
--
发表时间：
2006
期刊：
影响因子：
--
作者：
M. Köster;S. Turek
通讯作者：
S. Turek
DOI：
10.1051/m2an/1996300202371
发表时间：
1992
期刊：
Forschungsberichte, TU Munich
影响因子：
--
作者：
R. Hoppe;B. Wohlmuth
通讯作者：
R. Hoppe;B. Wohlmuth
求解三维非相容有限元不可压缩纳维斯托克斯方程的并行多重网格算法
DOI：
--
发表时间：
1996
期刊：
Conference on Parallel Computational Fluid Dynamics
影响因子：
--
作者：
H. Oswald;S. Turek
通讯作者：
S. Turek
DOI：
--
发表时间：
1988
期刊：
--
影响因子：
--
作者：
S. C. Brenner
通讯作者：
S. C. Brenner