The Virtual Element Method For Poroelasticity

The poroelasticity model describes the interaction between the fluid flow and deforma-tion in an elastic porous material.In this model,the motion of the fluid is described by Darcy's law,and the deformation of the porous medium is governed by linear elasticity.It was first proposed by Terzaghii and then summarized and extended by Biot.Biot's poroelastic model has been used in a diverse range of engineering fields,including reser-voir engineering,biomechanics,and materials science.Because of the complex coupled nature of the equations,computational simulations become essential to solving poroe-lastic problems.At first,people only considered displacement and pressure as unknown variables in the two-field model.The finite element method is the most common to deal with this problem.Then,more researchers began to consider the three-field model with the unknown of displacement,velocity and pressure.In this paper,we combine the virtual element method with the poroelasticity so that on the arbitrary model,the numerical solution of the model can be calculated and satisfied the convergence using appropriate degrees of freedom and the projection operators,and without calcuating the internal basis functions.The following is our studying of time-independent poroelasticity:(?)In the first part of this paper,we introduce the research background and domestic and foreign research summary of this problem.In the second part,the physical background and simplified form of this model are introduced briefly,and a variant of the continuous problem is constructed.We obtain the mixed virtual element method for poroelasticity through the construction of space and discrete bilinear form,the choice of freedom in chapter 3,and the existence and uniqueness of the constructed virtual element format solution are obtained.Then the optimal error estimates of displacement,pressure and velocity are obtained.At the end of the paper,several simple numerical examples are applied to verify that,the experimental results are consistent with the theoretical analysis results.