Time-Decoupled High Order Continuous Space-Time Finite Element Schemes for the Heat Equation
Time-Decoupled High Order Continuous Space-Time Finite Element Schemes for the Heat Equation
复制标题
热方程的时间解耦高阶连续时空有限元方案
DOI：
10.1137/130914589
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发表时间：
2014
影响因子：
3.1
通讯作者：
Kruse C
中科院分区：
文献类型：
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作者：
Kruse C
在第一版。方法:。动力机械。Engrg。Werder等人证明，通过时间不连续的Galerkin有限元方法对热方程的时间离散化可以通过对角化时间“Gram矩阵”来解耦。在本文中，我们提出了一种用连续伽辽金时间离散来求解热方程的伴随方法。因此，如果在时间上使用d次的分段多项式作为试函数，并且空间离散产生维数为M的系统，那么解耦后需要求解的是d个大小为M的系统，而不是单个大小为emd的系统。这些解耦系统需要复杂的算法，就像Werder等人的技术一样，但可以在现代多核架构上并行解决。我们使用Galerkin和谱元空间离散对三种不同的模型测试问题给出了时间多项式度高达6的数值测试，并证明了收敛率和时间超收敛率符合Aziz和Monk给出的边界。比较52:186(1989)，第255-274页。我们还将误差解释为计算时间的函数，并看到我们的高阶方案在单位计算时间的精度方面可能提供比Crank-Nicolson方法更高的效率——尽管在多核世界中，使用高度调整的迭代求解器，人们必须谨慎对待这种说法。最后，我们推测这些思想在不可压缩流体的Navier-Stokes方程中的应用。
In Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6685—6708 Werder et al. demonstrated that time discretizations of the heat equation by a temporally discontinuous Galerkin finite element method could be decoupled by diagonalising the temporal ‘Gram matrices’. In this article we propose a companion approach for the heat equation by using a continuous Galerkin time discretization. As a result, if piecewise polynomials of degree d are used as the trial functions in time and the spatial discretization produces systems of dimension M then, after decoupling, d systems of size M need to be solved rather than a single system of sizeMd. These decoupled systems require complex arithmetic, as did Werder et al.’s technique, but are amenable to parallel solution on modern multi-core architectures. We give numerical tests for temporal polynomial degrees up to six for three different model test problems, using both Galerkin and spectral element spatial discretizations, and show convergence and temporal superconvergence rates that accord with the bounds given by Aziz and Monk, Math. Comp. 52:186 (1989), pp. 255—274. We also interpret error as a function of computational time and see that our high order schemes may offer greater efficiency that the Crank-Nicolson method in terms of accuracy per unit of computational time—although in a multi-core world, with highly tuned iterative solvers, one has to be cautious with such claims. We close with a speculation on the application of these ideas to the Navier-Stokes equations for incompressible fluids.
高斯和高斯-洛巴托求积规则对时域四边形有限元方法精度的影响
DOI：
--
发表时间：
2009
期刊：
影响因子：
--
作者：
M. Duruflé;P. Grob;P. Joly
通讯作者：
P. Joly