We present an algorithm for the solution of a simultaneous space-time discretization of linear parabolic evolution equations with a symmetric differential operator in space. Building on earlier work, we recast this discretization into a Schur-complement equation whose solution is a quasi-optimal approximation to the weak solution of the equation at hand. Choosing a tensor-product discretization, we arrive at a remarkably simple linear system. Using wavelets in time and standard finite elements in space, we solve the resulting system in linear complexity on a single processor, and in polylogarithmic complexity when parallelized in both space and time. We complement these theoretical findings with large-scale parallel computations showing the effectiveness of the method.
翻译:我们提出了一个算法,用于解决线性抛物线进化方程式同时的时时分分化和空间对称分解操作器。基于先前的工作,我们将这种分解转换成Schur-compul化方程式,其解决办法是接近手边方方程式的微弱解决方案的准最佳近似值。我们选择了一种高压产品分解,我们到达了一个非常简单的线性系统。使用时序波和标准的空间有限元素,我们用单一的处理器和时间平行的多元复杂度来解决由此形成的系统线性复杂度。我们用显示方法有效性的大规模平行计算来补充这些理论结论。