On a finite time interval $(0,T)$, we consider the multiresolution Galerkin discretization of a modified Hilbert transform $\mathcal H_T$ which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $\geq 1$ with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces $ H^{s}_{0,}(0,T)$ and $ H^{s}_{,0}(0,T)$. These bases provide multilevel splittings of the temporal discretization spaces into "increment" or "detail" spaces of direct sum type. Via algebraic tensor-products of these temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases), sparse space-time tensor-product spaces are obtained, which afford a substantial reduction in the number of the degrees of freedom as compared to time-marching discretizations. In addition, temporal spline-wavelet bases allow to compress certain nonlocal integrodifferential operators which appear in stable space-time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. An efficient preconditioner is proposed that affords linear complexity solves of the linear system of equations which results from the sparse space-time Galerkin discretization.

翻译：在有限时间区间$(0,T)$上，我们考虑线性扩散方程时空伽辽金离散中出现的修正希尔伯特变换$\mathcal H_T$的多分辨伽辽金离散。为此，我们在$(0,T)$上设计由分段多项式构成的样条-小波系统，多项式次数$\geq 1$并具有足够多的消失矩，这些系统构成索博列夫空间$ H^{s}_{0,}(0,T)$和$ H^{s}_{,0}(0,T)$中的里斯基。这些基提供了时间离散空间到直和型"增量"或"细节"空间的多层分解。通过将这些时间多层离散与空间域中的标准分层有限元空间（使用标准拉格朗日有限元基）进行代数张量积，得到稀疏时空张量积空间，与时间推进离散相比，该空间大幅减少了自由度数。此外，时间样条-小波基能够压缩出现在初边值问题稳定时空变分公式中的某些非局部积分微分算子，例如热方程和声波方程。本文提出一种高效预处理器，可实现对稀疏时空伽辽金离散所得线性方程组的线性复杂度求解。