Resolvent Krylov subspace approximation to operator functions
Resolvent Krylov subspace approximation to operator functions
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DOI：
10.1007/s10543-011-0367-8
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发表时间：
2012-09
影响因子：
1.5
通讯作者：
Volker Grimm
中科院分区：
文献类型：
--
作者：
Volker Grimm
我们考虑算子函数在预解Krylov子空间中的逼近。除了许多其他的应用外，这种近似对于用指数积分器数值求解发展方程时出现的φ函数的逼近也很有兴趣。众所周知，无指数衰减矩阵函数的Krylov子空间方法在步长大于算子范数时表现出超线性收敛行为。因此，对于无界算子，Krylov近似可能不收敛。在这篇文章中，我们分析了一种有理Krylov子空间方法，它不仅对有限元或有限差分逼近收敛到微分算子，而且对值域位于左半平面上的抽象的无界算子也收敛。与标准的Krylov方法不同，收敛与离散化算子的范数无关，从而与空间离散化无关。我们将讨论有限元离散的有效实现，并用数值实验来说明我们的分析。
We consider the approximation of operator functions in resolvent Krylov subspaces. Besides many other applications, such approximations are currently of high interest for the approximation ofφ-functions that arise in the numerical solution of evolution equations by exponential integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behaviour if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we analyse a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators whose field of values lies in the left half plane. In contrast to standard Krylov methods, the convergence will be independent of the norm of the discretised operator and thus of the spatial discretisation. We will discuss efficient implementations for finite element discretisations and illustrate our analysis with numerical experiments.