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：本文提出了一系列用于求解孔隙弹性或热弹性耦合双曲-抛物型系统的时空有限元方法（STFEMs）。证明了离散问题的良态性。通过自然嵌入方式，可在计算可行的网格上实现继承连续问题解丰富结构的高阶近似。然而，该类方法中不同成员的代数系统具有复杂的块结构及求解特性。我们提出并分析了一种用于GMRES迭代的鲁棒几何多重网格（GMG）预处理器。该GMG方法采用局部Vanka型平滑器，其作用以精确数学方式定义。由于未知量存在非局部耦合机制，平滑器在单元片上执行，从而确保误差频率的衰减。通过一系列数值实验（包括具有实际挑战性的三维基准算例），验证并确认了该求解器在STFEMs中的高效性。分析了其并行可扩展性。除传统性能工程研究外，本文还从算法设计、调优及其硬件实现的附加新兴维度出发，探究了求解器的能效特性。