An eigen-based high-order expansion basis for structured spectral elements
An eigen-based high-order expansion basis for structured spectral elements
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DOI：
10.1016/j.jcp.2011.08.009
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发表时间：
2011-09
期刊：
影响因子：
--
通讯作者：
X. Zheng;S. Dong
中科院分区：
文献类型：
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作者：
X. Zheng;S. Dong
我们提出了一个基于特征的高阶展开基的谱元方法与结构元素。新的基础表现出显着上级的数值效率，在系数矩阵和共轭梯度求解器的收敛迭代次数的条件，常用的Jacobi多项式为基础的扩展基础。这个基础的结果在非常稀疏的质量矩阵，它是非常适合对角预处理。大量的数值实验表明，与新的基础和一个简单的对角预条件的共轭梯度迭代收敛的数量基本上没有依赖或只有一个很弱的依赖于元素的顺序。展开基由一组特殊的一维（1D）基函数的张量积构成。一维内部模式的构造，使内部的质量和刚度矩阵同时对角，并具有相同的条件数。1D顶点模式被构造为与所有内部模式正交。研究了新基础的性能，并与其他扩展基础进行了比较。
We present an eigen-based high-order expansion basis for the spectral element approach with structured elements. The new basis exhibits a numerical efficiency significantly superior, in terms of the conditioning of coefficient matrices and the number of iterations to convergence for the conjugate gradient solver, to the commonly-used Jacobi polynomial-based expansion basis. This basis results in extremely sparse mass matrices, and it is very amenable to the diagonal preconditioning. Ample numerical experiments demonstrate that with the new basis and a simple diagonal preconditioner the number of conjugate gradient iterations to convergence has essentially no dependence or only a very weak dependence on the element order. The expansion bases are constructed by a tensor product of a set of special one-dimensional (1D) basis functions. The 1D interior modes are constructed such that the interior mass and stiffness matrices are simultaneously diagonal and have identical condition numbers. The 1D vertex modes are constructed to be orthogonal to all the interior modes. The performance of the new basis has been investigated and compared with other expansion bases.