# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 1 Introduction
[PERSON] et al. [28] studied the causes of pressure oscillations in low-permeable and low-compressible media by presenting two, three and four field mixed formulations in terms of the field variables displacement, pore fluid pressure, fluid velocity and solid skeleton stress. A posteriori error estimation and adaptive refinement in poroelasticity, as a numerical stabilization technique for ill-conditioning or solution oscillations, has been studied by very few researchers. [PERSON] and [PERSON] [29] presented a novel approach for space-time adaptive finite element analysis for the coupled consolidation problem in geomechanics. [PERSON] and [PERSON] [30] proposed an a posteriori error estimator for plane-strain geotechnical analyses based on superconvergent patch recovery with application to [PERSON]'s consolidation problem. They later extended the application of the a posteriori estimator for axisymmetric geotechnical analyses in [31]. Adaptive isogeometric finite element analysis using LR B-Splines [32] and recovery based error estimators [33], was presented for steady-state groundwater flow problems by [PERSON] et al. [34]. Isogeometric finite element analysis of poroelasticity was first presented by [PERSON] et al. [35]. The advantages of the smoothness of the basis functions in isogeometric analysis were highlighted in their application. One of the advantages of higher continuity is that the numerical implementation results in a locally mass conserving flow between knotspans, analogous to elements in finite element analysis. However, the formulation presented relied on equal orders of interpolation for the field variables in poroelasticity, namely displacement and pore fluid pressure. Such a formulation, while still useful for several applications without significant numerical challenges, has limitations when it comes to problems where the material properties or boundary conditions are problematic. Mixed finite element and NURBS-based isogeometric analysis for Darcy and Darcy-Brinkman flow through deformable porous media was presented in [36]. Other related studies include isogeometric analysis of fluid-saturated porous media including flow in the cracks [37], mixed isogeometric analysis of strongly coupled diffusion in porous materials [38] and isogeometric analysis of fracture propagation in saturated porous media due to a pressurised non-Newtonian fluid [39]. Regarding mixed methods for isogeometric finite elements we want to mention the use of so-called Mixed Integration Point (MIP) recently proposed in [40] to achieve the advantages of mixed formulation for avoiding locking in isogeometric analysis of thin shell problems [41; 42] without explicitly introducing the mixed set of variables. In this paper, we present mixed formulation for isogeometric analysis of poroelasticity. We demonstrate through numerical examples that mixed isogeometric analysis can overcome pressure oscillations with the right choice of continuity and meshing, without introducing additional field variables in the mathematical formulation. The results are discussed in comparison with related numerical studies The paper is structured as follows. In Section 2, the governing equations of poroelasticity are presented. The fundamentals of isogeometric analysis and its particular features of interest within the current context are discussed in Section 3. Numerical examples are given in Section 4 and the observations are summarized with concluding remarks in Section 5.