Convergence Analysis of Spatially Adaptive Rothe Methods
Convergence Analysis of Spatially Adaptive Rothe Methods
复制标题
空间自适应Rothe方法的收敛性分析
DOI：
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发表时间：
2014
影响因子：
3
通讯作者：
R. Schilling
中科院分区：
文献类型：
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作者：
P. A. Cioica;S. Dahlke;N. Döhring;U. Friedrich;S. Kinzel;F. Lindner;T. Raasch;K. Ritter;R. Schilling
本文研究的是抛物型演化方程水平直线法的收敛性分析。通过 $$S$$S 级一步法进行时间半离散化后，每个时间步长的椭圆级方程用自适应空间离散化方案求解。我们研究了如何调整每个时间步长的容差，以便在存在空间离散误差的情况下也保持时间步长的渐近时间收敛顺序。特别是，我们讨论了线性隐式时间积分器和空间中的自适应小波离散化的情况。使用偏微分方程正则理论和非线性逼近理论的概念，我们确定了整体方案的自由度上限，该上限是自适应地将解逼近到指定公差所需的。
This paper is concerned with the convergence analysis of the horizontal method of lines for evolution equations of the parabolic type. Following a semidiscretization in time by $$S$$S-stage one-step methods, the resulting elliptic stage equations per time step are solved with adaptive space discretization schemes. We investigate how the tolerances in each time step must be tuned in order to preserve the asymptotic temporal convergence order of the time stepping also in the presence of spatial discretization errors. In particular, we discuss the case of linearly implicit time integrators and adaptive wavelet discretizations in space. Using concepts from regularity theory for partial differential equations and from nonlinear approximation theory, we determine an upper bound for the degrees of freedom for the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance.