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In this paper we develop adaptive iterative coupling schemes for the Biot system modeling coupled poromechanics problems. We particularly consider the space-time formulation of the fixed-stress iterative scheme, in which we first solve the…

Numerical Analysis · Mathematics 2024-12-20 Elyes Ahmed , Jan Martin Nordbotten , Florin Adrian Radu

We develop non-overlapping domain decomposition methods for the Biot system of poroelasticity in a mixed form. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a…

Numerical Analysis · Mathematics 2021-08-04 Manu Jayadharan , Eldar Khattatov , Ivan Yotov

A semi-implicit two-phase double-point Material Point Method (MPM) formulation, based on the incremental fractional-step method to model large deformation geotechnical problems has been derived. The semi-implicit formulation has two…

Numerical Analysis · Mathematics 2025-08-22 Mian Xie , Pedro Navas , Susana Lopez-Querol

In this paper, we propose a new multiphysics finite element method for a Biot model with secondary consolidation in soil dynamics. To better describe the processes of deformation and diffusion underlying in the original model, we…

Numerical Analysis · Mathematics 2022-04-08 Zhihao Ge , Wenlong He

In this paper we present and analyze a fully-mixed formulation for the coupled problem arising in the interaction between a free fluid and a flow in a poroelastic medium. The flows are governed by the Stokes and Biot equations,…

Numerical Analysis · Mathematics 2021-05-25 Sergio Caucao , Tongtong Li , Ivan Yotov

This paper presents a novel parallel splitting algorithm for solving quasi-static multiple-network poroelasticity (MPET) equations. By introducing a total pressure variable, the MPET system can be reformulated into a coupled…

Numerical Analysis · Mathematics 2025-07-29 Jijing Zhao , Huangxin Chen , Mingchao Cai , Shuyu Sun

We consider the quasi-static Biot's consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement $\boldsymbol{u}$ of the solid matrix, the seepage velocity $\boldsymbol{v}$…

Numerical Analysis · Mathematics 2021-07-07 Johannes Kraus , Philip L. Lederer , Maria Lymbery , Joachim Schöberl

In this paper, we present two multiple scalar auxiliary variable (MSAV)-based, finite element numerical schemes for the Abels-Garcke-Gr{\"u}n (AGG) model, which is a thermodynamically consistent phase field model of two-phase incompressible…

Numerical Analysis · Mathematics 2024-08-09 Jiancheng Wang , Maojun Li , Cheng Wang

In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of…

Numerical Analysis · Mathematics 2025-10-23 Ruben Caraballo , Chansophea Wathanak In , Alberto F. Martín , Ricardo Ruiz-Baier

This paper studies the Craig variant of the Golub-Kahan bidiagonalization algorithm as an iterative solver for linear systems with saddle point structure. Such symmetric indefinite systems in 2x2 block form arise in many applications, but…

Computational Engineering, Finance, and Science · Computer Science 2018-08-24 Mario Arioli , Carola Kruse , Ulrich Ruede , Nicolas Tardieu

We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element method for Darcy flow, the method utilizes…

Numerical Analysis · Mathematics 2019-07-30 Ilona Ambartsumyan , Eldar Khattatov , Jan M. Nordbotten , Ivan Yotov

This paper is devoted to the stability analysis of a classical three-field formulation of Biot's consolidation model where the unknown variables are the displacements, fluid flux (Darcy velocity), and pore pressure. Specific…

Numerical Analysis · Mathematics 2018-06-21 Qingguo Hong , Johannes Kraus

This paper studies the thermo-poroelasticity model. By introducing an intermediate variable, we transform the original three-field model into a four-field model. Building upon this four-field model, we present both a coupled finite element…

Numerical Analysis · Mathematics 2025-07-23 Mingchao Cai , Jingzhi Li , Ziliang Li , Qiang Liu

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum…

Numerical Analysis · Mathematics 2019-08-02 Denis Spiridonov , Maria Vasilyeva , Eric T. Chung

In this paper, we extend the inexact Uzawa algorithm in [Q. Hu, J. Zou, SIAM J. Matrix Anal., 23(2001), pp. 317-338] to the nonsymmetric generalized saddle point problem. The techniques used here are similar to those in [Bramble \emph{et…

Numerical Analysis · Mathematics 2014-08-26 Hailun Shen , Hua Xiang

In the paper, we introduce several accelerate iterative algorithms for solving the multiple-set split common fixed-point problem of quasi-nonexpansive operators in real Hilbert space. Based on primal-dual method, we construct several…

Optimization and Control · Mathematics 2023-06-08 Chenzheng Guo , Jing Zhao

We consider a non-linear extension of Biot's model for poromechanics, wherein both the fluid flow and mechanical deformation are allowed to be non-linear. We perform an implicit discretization in time (backward Euler) and propose two…

Numerical Analysis · Mathematics 2017-02-02 Manuel Borregales , Florin A. Radu , Kundan Kumar , Jan M. Nordbotten

We consider the systematic numerical approximation of Biot's quasistatic model for the consolidation of a poroelastic medium. Various discretization schemes have been analysed for this problem and inf-sup stable finite elements have been…

Numerical Analysis · Mathematics 2020-01-01 Herbert Egger , Mania Sabouri

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of…

Numerical Analysis · Mathematics 2021-03-24 Martin Neumuller , Iain Smears

We develop a novel fast iterative moment method for the steady-state simulation of near-continuum flows, which are modeled by the high-order moment system derived from the Boltzmann-BGK equation. The fast convergence of the present method…

Numerical Analysis · Mathematics 2025-07-30 Guanghan Li , Chunwu Wang , Zhicheng Hu