We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modeling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in time and inf-sup stable pairs of finite element spaces for the spatial variables are investigated. Optimal order error estimates are proved by an analysis in weighted norms that depict the energy of the system's unknowns. A further important ingredient and challenge of the analysis is the control of the couplings terms. The techniques developed here can be generalized to other families of Galerkin space discretizations and advanced models. The error estimates are confirmed by numerical experiments, also for higher order piecewise polynomials in time and space. The latter lead to algebraic systems with complex block structure and put a facet of challenge on the design of iterative solvers. An efficient solution technique is referenced.

翻译：本文研究了一类耦合双曲弹性动力学与抛物输运过程的物理耦合系统（模拟孔隙弹性与热弹性）的时空有限元数值逼近。将方程重写为时间一阶系统，探讨了时间方向的连续伽辽金离散格式与空间变量的inf-sup稳定有限元空间对。通过刻画系统未知量能量的加权范数分析，证明了最优阶误差估计。分析中另一个关键要素与挑战在于耦合项的控制。本文发展的技术可推广至其他伽辽金空间离散族及高阶模型。数值实验验证了误差估计的可靠性，并针对时空高阶分片多项式进行了测试——后者产生具有复杂块结构的代数系统，给迭代求解器设计带来了新的挑战。文中引述了一种高效求解技术。