This paper concerns monolithic and splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE’s, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic displacement. Due to the presence of a nonlinear convective transport term in the energy balance equation, it is convenient to have access to both the pressure and temperature gradients. Hence, we introduce these as two additional variables and extend the original three-field model to a five-field model. For the numerical solution of this five-field formulation, we compare six approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from the mechanics, suitable for varying coupling strength between the three physical processes. The approaches have in common a simultaneous application of the so-called $L$-scheme, which works both to stabilize iterative splitting as well as to linearize nonlinear problems, and can be seen as a generalization of the Undrained and Fixed-Stress Split algorithms. More precisely, the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion. We provide a convergence proof for the derived algorithms, and demonstrate their performance through several numerical examples investigating different strengths of the coupling between the different processes.

本文涉及耦合非线性热多孔弹性模型问题的基于整体和基于分裂的迭代过程。我们考虑的热多孔弹性模型问题被公式化为PDE的三场系统，它由一个能量平衡方程，一个质量平衡方程和一个动量平衡方程组成，其中主要变量是温度，流体压力和弹性位移。由于能量平衡方程中存在非线性对流输运项，因此可以方便地访问压力和温度梯度。因此，我们将它们作为两个附加变量引入，并将原始的三场模型扩展到五场模型。对于此五字段公式的数值解，我们比较了六种不同的方法，这些方法的不同之处在于我们如何处理流和/或热量和/或力学之间的耦合/去耦，适合改变三个物理过程之间的耦合强度。这些方法通常同时应用所谓的$\mathrm{大号}$-scheme既可以稳定迭代拆分，也可以使非线性问题线性化，并且可以看作是不排水和固定应力拆分算法的推广。更准确地说，导出的过程将非线性且完全耦合的问题转换为一组较简单的子问题，这些子问题将以迭代方式依次解决。我们为派生的算法提供了收敛证明，并通过调查几个不同过程之间的不同耦合强度的几个数值示例来证明它们的性能。