论文标题
抛物线问题的最佳局部近似空间
Optimal local approximation spaces for parabolic problems
论文作者
论文摘要
我们提出了当地时空近似空间，以解决抛物线问题，这些问题在Kolmogorov的意义上是最佳的，并且可以用于多尺度和域分解方法中。扩散系数在空间和时间上可以任意粗糙。为了构建局部近似空间，我们考虑了一个紧凑的转移操作员，该操作员作用于本地解决方案的空间，并覆盖全日制尺寸。然后，最佳局部空间由转移操作员的左单数向量给出。为了证明后者的紧凑性，我们将合适的抛物线片不平等与Aubin-Lions的紧凑定理相结合。与椭圆形相反[I. Babuška和R. Lipton，多尺度模型。 Simul。，9（2011），第373-406页]我们需要附加的规律性结果来结合两个结果。此外，我们采用了广义有限元方法来对局部空间进行融合并构建全局解决方案的近似值。由于我们的方法会减少时空碱基，因此全局近似的计算不需要时间步进方法，因此在计算上是有效的。此外，我们得出了严格的本地和全局一个先验错误界限。详细说明，我们通过$ l^2（h^1）$ - 标准中的本地错误限制了全局近似错误，并指出传输操作员映射到达此规范的空间。数值实验证明了转移操作员的单数值的指数衰减以及局部和全局近似误差，即有关空间和时间的高对比度或多尺度结构的问题。
We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in space and time. To construct local approximation spaces we consider a compact transfer operator that acts on the space of local solutions and covers the full time dimension. The optimal local spaces are then given by the left singular vectors of the transfer operator. To prove compactness of the latter we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin-Lions. In contrast to the elliptic setting [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406] we need an additional regularity result to combine the two results. Furthermore, we employ the generalized finite element method to couple local spaces and construct an approximation of the global solution. Since our approach yields reduced space-time bases, the computation of the global approximation does not require a time stepping method and is thus computationally efficient. Moreover, we derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the $L^2(H^1)$-norm, noting that the space the transfer operator maps to is equipped with this norm. Numerical experiments demonstrate an exponential decay of the singular values of the transfer operator and the local and global approximation errors for problems with high contrast or multiscale structure regarding space and time.
