Article
Mathematics, Applied
Feng Wang, Mingchao Cai, Gang Wang, Yuping Zeng
Summary: This paper presents a computational method for solving the poroelasticity problem using weak virtual element method. The approach discretizes the flux velocity, pressure, and elastic displacement, and provides a fully discrete scheme. Numerical experiments are conducted to validate the efficacy of the method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Engineering, Multidisciplinary
Wenlong He, Zhihao Ge
Summary: This paper proposes a new mixed finite element method for a swelling clay model with secondary consolidation. The original model is reformulated to better describe the multi-physics processes by introducing a new variable. The existence and uniqueness of the weak solution are proved through PDE analysis. A fully discrete time-stepping scheme is designed using mixed finite element method with P-2-P-1-P-1 element pairs for the space variables and backward Euler method for the time variable. Stability analysis and optimal convergence order error estimates are provided. Numerical examples are presented to verify the theoretical results and illustrate the significant effect of the secondary consolidation term on displacement.
APPLIED MATHEMATICAL MODELLING
(2022)
Article
Engineering, Multidisciplinary
Peter Hansbo, Mats G. Larson
Summary: In this paper, a nonconforming rotated bilinear tetrahedral element is applied to the Stokes problem in R-3, demonstrating stability in combination with a piecewise linear, continuous approximation of the pressure. This element provides an approximation similar to the well-known Taylor-Hood element but with fewer degrees of freedom, and fulfills Korn's inequality, ensuring stability even when the Stokes equations are written on stress form for use in free surface flow.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Hongpeng Li, Hongxing Rui
Summary: In this paper, the Biot's consolidation model and Darcy-Forchheimer equation are used to study the relationship between fluid velocity and pressure. The model is nonlinear and involves the unknown variables of solid displacement, fluid velocity, and pressure. The paper presents a stable method for solving this problem and provides error estimates for the finite element approximations. Numerical experiments are conducted to validate the theoretical analysis and investigate pressure variations in poroelasticity problems.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Yuping Zeng, Mingchao Cai, Liuqiang Zhong
Summary: A mixed finite element method is proposed for the Biot consolidation problem in poroelasticity. Approximations are used for displacement and fluid pressure using Crouzeix-Raviart nonconforming finite elements and node conforming finite elements, respectively. The well-posedness of the fully discrete scheme is established, and an a priori error estimate with optimal order in the energy norm is derived. Numerical experiments are conducted to validate the theoretical results.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Namshad Thekkethil, Simone Rossi, Hao Gao, Scott I. Heath Richardson, Boyce E. Griffith, Xiaoyu Luo
Summary: This paper proposes a variational multiscale method to stabilize a linear finite element method for nonlinear poroelasticity. The method is suitable for implicit time integration of anisotropic and incompressible poroelastic formulations. A detailed numerical methodology is presented for a monolithic formulation that includes both structural dynamics and Darcy flow. The method is verified using benchmark cases and demonstrated to have second-order accuracy.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Yuxiang Chen, Zhihao Ge
Summary: In this paper, a multiphysics finite element method is proposed for the quasi-static thermoporoelasticity model with small Peclet number. The method is proved to have existence, uniqueness, stability and optimal convergence order. Numerical examples are provided to verify the theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Zhihao Ge, Shuaichao Pei, Yinyin Yuan
Summary: This paper proposes a two-level multiphysics finite element method for a nonlinear poroelasticity model. The original fluid-solid coupled problem is reformulated into a fluid-fluid coupled problem to reveal the multiphysics processes and overcome the locking phenomenon. Two fully discrete coupling and decoupling methods are proposed to reduce the computational cost.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Jeonghun J. Lee
Summary: This paper discusses the finite element discretization and a priori error analysis of low-frequency dynamic poroelasticity models. Unlike the widely used quasi-static poroelasticity models, the dynamic models are hyperbolic partial differential equations with acceleration terms of solid and fluid phases. The problem is reformulated as a symmetric hyperbolic system and discretized using two mixed finite element methods. The error analysis of semidiscrete solutions is discussed in detail, and numerical results of the backward Euler fully discrete scheme are presented.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Aycil Cesmelioglu, Jeonghun J. Lee, Sander Rhebergen
Summary: We propose an embedded-hybridizable discontinuous Galerkin finite element method for the total pressure formulation of the quasi-static poroelasticity model. By approximating the displacement and the Darcy velocity using discontinuous piece-wise polynomials and enforcing H(div)-conformity using Lagrange multipliers, we ensure stability of the semi-discrete problem and well-posedness of the fully discrete problem. Additionally, space-time a priori error estimates are derived and numerical examples confirm that the proposed discretization is free of volumetric locking.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Liu Qian, Shi Dongyang
Summary: This paper develops a nonconforming mixed finite element method for the time-dependent Navier-Stokes problem with a nonlinear damping term. Superconvergent estimates for the velocity and pressure are rigorously deduced through the use of certain techniques, and global superconvergent results are obtained through postprocessing. Numerical results are provided to confirm the theoretical analysis.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Lingling Sun, Yidu Yang
Summary: This paper discusses the a posteriori error estimates and adaptive algorithm of non-conforming mixed finite elements for the Stokes eigenvalue problem. The reliability and efficiency of the error estimators are proven. Two adaptive algorithms, direct AFEM and shifted-inverse AFEM, are built based on the error estimators. Numerical experiments and theoretical analysis show that the numerical eigenvalues obtained by these algorithms achieve optimal convergence order and approximate the exact solutions from below.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Yujie Liu, Junping Wang
Summary: An extended P-1 nonconforming finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations on general polytopal partitions, inspired by the simplified weak Galerkin method. The method reduces computational complexity by utilizing only the degrees of freedom on the boundary of each element. Numerical stability and optimal order of error estimates in H-1 and L-2 norms are established for the numerical solutions, with a superconvergence phenomenon noted on rectangular partitions through numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Hongpeng Li, Hongxing Rui
Summary: This article investigates the three-field Biot's consolidation model with the consideration of Darcy-Forchheimer law for fluid velocity. The linear finite element and RT mixed finite element are proposed to approximate the unknown variables, and error estimates are provided for both semidiscrete and fully discrete schemes. Numerical tests confirm the accuracy of the P1-RT0-P0 method under positive and null constraint storage coefficients c0.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Zhihao Ge, Jin'ge Pang, Jiwei Cao
Summary: This article proposes a multiphysics mixed finite element method with Nitsche's technique for solving the Stokes-poroelasticity problem. The method reformulates the problem by introducing pseudo-pressures and uses classical stable finite element pairs to deal with it conveniently. A loosely-coupled time-stepping method is proposed to solve the subproblems at each time step, and the method does not require any restriction on the choice of discrete approximation spaces on each side of the interface. The stability analysis and error estimates of the method are provided, and numerical tests show its good stability and absence of locking phenomenon.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)