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We study numerical algorithms for solving Biot’s model. Based on a three-field reformulation, we present some algorithms that are inspired by the work of Chaabane et al. (Comput MathAppl 75(7):2328–2337) and Lee (Unconditionally stable second order convergent partitioned methods for multiple-network poroelasticity arXiv:1901.06078, 2019) for decoupling the computation of Biot’s model. A new theoretical framework is developed to analyze the algorithms. Considering a uniform temporal discretization, these algorithms solve the coupled model on the first time level. On the remaining time levels, one algorithm solves a reaction-diffusion subproblem first and then solves a generalized Stokes subproblem.Another algorithm reverses the order of solving the two subproblems. Our algorithms manage to decouple the numerical computation of the coupled system while retaining the convergence properties of the original coupled algorithm. Theoretical analysis is conducted to show that these algorithms are unconditionally stable and optimally convergent.Numerical experiments are also carried out to validate the theoretical analysis and demonstrate the advantages of the proposed algorithms. 

Biot's consolidation model in poroelasticity describes the interaction between the fluid and the deformable porous structure. Based on the fixed-stress splitting iterative method proposed by Mikelic et al. (Computat Geosci, 2013), we present a network approach to solve Biot's consolidation model using physics-informed neural networks (PINNs). Methods Two independent and small neural networks are used to solve the displacement and pressure variables separately. Accordingly, separate loss functions are proposed, and the fixed stress splitting iterative algorithm is used to couple these variables. Error analysis is provided to support the capability of the proposed fixed-stress splitting-based PINNs (FS-PINNs). Results Several numerical experiments are performed to evaluate the effectiveness and accuracy of our approach, including the pure Dirichlet problem, the mixed partial Neumann and partial Dirichlet problem, and the Barry-Mercer's problem. The performance of FS-PINNs is superior to traditional PINNs, demonstrating the effectiveness of our approach. Discussion Our study highlights the successful application of PINNs with the fixed-stress splitting iterative method to tackle Biot's model. The ability to use independent neural networks for displacement and pressure offers computational advantages while maintaining accuracy. The proposed approach shows promising potential for solving other similar geoscientific problems. 

This paper is concerned with numerical algorithms for Biot model. By introducing an intermediate variable, the classical 2-field Biot model is written into a 3-field formulation. Based on such a 3-field formulation, we propose a coupled algorithm, some time-extrapolation based decoupled algorithms, and an iterative decoupled algorithm. Our focus is the analysis of the iterative decoupled algorithm. It is shown that the convergence of the iterative decoupled algorithm requires no extra assumptions on physical parameters or stabilization parameters. Numerical experiments are provided to demonstrate the accuracy and efficiency of the proposed method. 

In this paper, we develop parameter-robust numerical algorithms for Biot model and apply the algorithms in brain edema simulations. By introducing an intermediate variable, we derive a multiphysics reformulation of the Biot model. Based on the reformulation, the Biot model is viewed as a generalized Stokes subproblem combining with a reaction–diffusion subproblem. Solving the two subproblems together or separately leads to a coupled or a decoupled algorithm. We conduct extensive numerical experiments to show that the two algorithms are robust with respect to the key physical parameters. The algorithms are applied to study the brain swelling caused by abnormal accumulation of cerebrospinal fluid in injured areas. The effects of the key physical parameters on brain swelling are carefully investigated. It is observed that the permeability has the biggest influence on intracranial pressure (ICP) and tissue deformation; the Young’s modulus and the Poisson ratio do not affect the maximum value of ICP too much but have big influence on the tissue deformation and the developing speed of brain swelling. 

Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. Forthe posteriorierror estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.