An efficient solution algorithm for space–time finite element method
An efficient solution algorithm for space–time finite element method
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DOI：
10.1007/s00466-018-1603-8
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发表时间：
2018-07
影响因子：
4.1
通讯作者：
Rui Zhang;L. Wen;Jin-you Xiao;D. Qian
中科院分区：
文献类型：
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作者：
Rui Zhang;L. Wen;Jin-you Xiao;D. Qian
从时间不连续Galerkin(TDG)格式出发，提出了一种高效的时空有限元求解算法。该算法的特点是迭代求解器由一种新颖有效的预条件器加速。该预处理器是基于耦合空时系统矩阵的块结构构造的，该块结构表示为时间和空间子矩阵的Kronecker积的相加。通过这种独特的分解，迭代求解器中计算量最大的运算，即矩阵运算，随后利用Kronecker积的逆性质进行了优化和加速。理论分析和数值算例均表明，与已有的基于TDG的时空有限元实现方法相比，该算法具有更好的性能。它将求解时空方程的计算量降低到与常规有限元求解刚度方程相同的阶数，从而使时空有限元在工程上的实际应用成为可能。
An efficient solution algorithm has been developed for space–time finite element method that is derived from time discontinuous Galerkin (TDG) formulation. The proposed algorithm features an iterative solver accelerated by a novel and efficient preconditioner. This preconditioner is constructed based on the block structure of coupled space–time system matrix, which is expressed as addition of Kronecker products of temporal and spatial submatrices. With this unique decomposition, the most computationally intensive operations in the iterative solver, i.e. matrix operations, are subsequently optimized and accelerated employing the inverse property of Kronecker product. Theoretical analysis and numerical examples both demonstrate that the proposed algorithm provides significantly better performance than the already developed implementations for TDG-based space–time FEM. It reduces the computational cost of solving space–time equations to the same order of solving stiffness equations associated with regular FEM, thereby enabling practical implementation of the space–time FEM for engineering applications.