We present families of space-time finite element methods (STFEMs) for a coupled hyperbolic-parabolic system of poro- or thermoelasticity. Well-posedness of the discrete problems is proved. Higher order approximations inheriting most of the rich structure of solutions to the continuous problem on computationally feasible grids are naturally embedded. However, the block structure and solution of the algebraic systems become increasingly complex for these members of the families. We present and analyze a robust geometric multigrid (GMG) preconditioner for GMRES iterations. The GMG method uses a local Vanka-type smoother. Its action is defined in an exact mathematical way. Due to nonlocal coupling mechanisms of unknowns, the smoother is applied on patches of elements. This ensures the damping of error frequencies. In a sequence of numerical experiments, including a challenging three-dimensional benchmark of practical interest, the efficiency of the solver for STFEMs is illustrated and confirmed. Its parallel scalability is analyzed. Beyond this study of classical performance engineering, the solver's energy efficiency is investigated as an additional and emerging dimension in the design and tuning of algorithms and their implementation on the hardware.
翻译:我们为混合的双曲线或热弹性系统展示了空间-时间限制元素方法(STFEMs)的组合。 已经证明了离散问题的准确性。 在计算可行的电网中,大多数丰富的解决方案结构都是自然嵌入的。 但是,对于这些家庭成员来说,代数系统的区块结构和解决方案变得日益复杂。 我们为GMRES的迭代提供并分析一个强大的几何多格多格格(GMG)的先决条件。 GMG方法使用一个本地的Vanka型滑动器。 它的行动是以精确的数学方式界定的。 由于非本地的未知的混合机制, 将光滑应用在元素的补补补上。 这确保了误差频率的宽度。 在一系列数字实验中, 包括具有挑战性的实际兴趣的三维基准, 演示并确认了STFEMs解决方案的效率。 它的平行的可缩放性。 除了这项对古典性工程的研究外, 解析器的能量效率作为设计及新变的硬件操作和演算法的补充, 调查了它们是如何进行进一步的。</s>