Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation
Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation
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DOI：
10.1515/jnum.2011.003
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发表时间：
2011-05-01
影响因子：
3
通讯作者：
Turek, S.
中科院分区：
文献类型：
--
作者：
Hussain, S.;Schieweck, F.;Turek, S.
作为纯量抛物型偏微分方程的典型例子，我们讨论了应用于热方程的连续Galerkin-Petrov和不连续Galerkin时间离散的数值性质。对于空间离散，我们使用一般二维网格上的双二次四边形有限元。我们讨论的时间离散化以及有效的方法来解决所产生的块系统的实施方面。在这里，我们比较了预处理的BiCGStab求解器作为一个适应几何多重网格求解器的Krylov空间方法。只有多重网格方法的收敛性几乎与网格大小和时间步长无关，才能得到有效的求解过程。通过数值实验，我们比较了不同的时间离散精度和计算成本。
We discuss numerical properties of continuous Galerkin-Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as well as efficient methods for solving the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a Krylov space method with an adapted geometrical multigrid solver. Only the convergence of the multigrid method is almost independent of the mesh size and the time step leading to an efficient solution process. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs.