On the multi-level splitting of finite element spaces
On the multi-level splitting of finite element spaces
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DOI：
10.1007/bf01389538
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发表时间：
1986
影响因子：
2.1
通讯作者：
H. Yserentant
中科院分区：
文献类型：
--
作者：
H. Yserentant
本文分析了用有限元空间的层次基代替通常的节点基，用有限元方法离散二阶自伴和正定平面椭圆边值问题时所产生的刚度矩阵的条件数。我们证明了这种刚度矩阵的条件数表现为O((logκ)2)，其中κ是关于结点基的刚度矩阵的条件数。在网格大小均匀的三角剖分的情况下，这意味着相对于有限元空间的层次基的刚度矩阵具有与节点基相似的条件数。我们定理的证明既不需要连续问题的正则性，也不需要离散化问题的正则性。特别是，我们不需要所采用的三角剖分的准一致性。由于有限元函数关于层次基的表示可以非常容易和快速地转换成它关于节点基的表示，我们的结果意味着，如果使用层次基或相关的预条件过程，共轭梯度法只需要O(Logn)步和O(Nlogn)次计算机运算就可以将误差的能量范数减少给定的因子。给出了有限元空间和要解的离散线性问题的尺寸。
In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log κ)2) where κ is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizehthis means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving likeinstead offor a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n)steps andO(n log n)computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Herendenotes the dimension of the finite element space and of the discrete linear problem to be solved.