# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 1 Introduction
They showed that there is an accuracy condition in the finite element analysis of consolidation by using a critical time-step, below which oscillatory solutions are observed. The derived critical time-step is strictly valid for a one-dimensional case and a uniform finite element mesh. [PERSON] [13] analyzed the numerical errors in the analysis of consolidation by finite elements. It was shown that the use of a mixed formulation for the field variables helps in reducing the pore pressure oscillations but may not remove them entirely. They instead used Gauss point smoothing to eliminate the pore pressure oscillations. Special finite elements for the analysis of consolidation were proposed by [PERSON] et al. [14]. They presented \"singularity\" elements to model pore pressures in the vicinity of free-draining loaded surfaces immediately after application of loads. The elements were special in that they use special interpolation schemes which reflect the actual variation of the field variables. The finite element method became a well-established method for the analysis of problems in poroelasticity and the mathematical properties of the governing equations and the numerical solution were studied in a further great detail. [PERSON] and [PERSON] [15] presented numerical analysis and error estimates of finite element approximations of [PERSON]'s consolidation problem. They used a mixed formulation and improved the rates of convergence by using a sequential Galerkin Petrov-Galerkin post-processing technique. In a further study [16], they investigated the stability and convergence of finite elements approximations of poroelasticity. They derived decay functions showing that the pore pressure oscillations, arising from an unstable approximation of the incompressibility constraint on the initial conditions, decay in time. Finite element analysis of consolidation with automatic time-stepping and error control was presented by [PERSON] and [PERSON] [17,18]. Automatic time increments were selected such that the temporal discretization error in the displacements is close to a specified tolerance. [PERSON] et al. [19] studied the ill-conditioning of finite element poroelasticity equations with a focus on the instabilities that may affect the pore pressure solution. They claim that the origin of most instabilities is due to the assumption that, for initial conditions, the porous medium behaves as an incompressible medium if the pore fluid is incompressible. They also argue that oscillatory pore pressure solutions may not always be observed for very stiff and low permeable materials depending on the critical time step. [PERSON] et al. [20] studied the numerical performance of projection methods in finite element consolidation models. [PERSON] et al. [21] proposed a large time increment (LATIN) computational strategy for problems of poroelasticity to improve the efficiency of the finite element analysis. A finite element formulation to overcome spatial pore pressure oscillations caused by small time increments was proposed by [PERSON] et al. [22]. [PERSON] et al. [23] compared standard and mixed finite element methods for poroelasticity. In particular, [PERSON] and least-squares mixed finite element methods were compared. They claim that [PERSON]'s method is able to preserve steep pressure gradients but overestimates the effective stresses. On the other hand, a least-squares mixed method was noticed to have the advantage of direct approximation of the primary variables and explicit approximation of Neumann type boundary conditions but to be computationally more expensive. A mixed least-squares finite element method for poroelasticity was also proposed by [PERSON] et al. [24], claiming that pore pressure oscillations are eliminated for different temporal discretizations. A coupling of mixed and continuous Galerkin finite element methods for poroelasticity was investigated for continuous and discrete in time cases by [PERSON] and [PERSON] [25; 26]. They also studied a coupling of mixed and discontinuous Galerkin finite-element methods, [27].