We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method. Further, a one-step time integration scheme is used for the numerical solution of the arising system of ordinary differential equations. For the latter, the parareal method decomposing the time interval into subintervals is employed. It leads to parallel solution of smaller time-dependent problems. At each time slice a pseudostationary elliptic heat equation is solved by means of a domain decomposition method (DDM). In the 2 d , case we employ a nonoverlapping Schur complement method, while in the 1 d case an overlapping Schwarz DDM is employed. We document computational efficiency, as well as theoretical convergence rates of FEM semi-discretization schemes on numerical examples.

中文翻译：

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1 维和 2 维空间瞬态热方程的域分解方法与拟实数相结合

我们为一维和二维空间中的瞬态热方程提出了一种并行求解算法。该问题通过最低阶符合有限元方法在空间中离散化。此外，一步时间积分方案用于常微分方程组的数值解。对于后者，采用将时间间隔分解为子间隔的超现实方法。它导致较小的时间相关问题的并行解决方案。在每个时间片上，通过域分解方法 (DDM) 求解伪平稳椭圆热方程。在 2 d 情况下，我们采用非重叠 Schur 补码方法，而在 1 d 情况下，采用重叠 Schwarz DDM。我们记录了计算效率，