# Mixed Method for Isogeometric Analysis of Coupled Flow and Deformation in Poroelastic Media
## 4 Numerical Examples
### Low Permeability Layer
Simulations with a graded mesh are also performed for the different combination of polynomial degrees and interface continuities. The graded mesh is generated such that more elements are concentrated at the material interfaces. The results from this case are shown in Figure 9. The pressure oscillations in the equal order case improve significantly in this case compared to the results from uniform mesh refinement. However, the oscillations still occur throughout the low permeability layer. The equal order results for linear basis functions show a slightly strange behavior in that the oscillations are lesser within the low permeability layer than for higher order elements, but show slightly higher oscillations at the top material interface. The results are again better with a \(C^{0}\) continuity at the material interfaces. The mixed simulation results also improve with a graded mesh. Almost no oscillations are noticed for combinations of higher polynomial degrees and \(C^{0}\) continuities at the material interfaces. To compare our IGA analysis results with standard finite elements, we refer to the extensive finite element study on pressure oscillations presented in [28] where the low permeability layer problem is one of the numerical examples investigated. [PERSON] et al. presented a systematic case study where they considered three different formulations for the mathematical model of poroelasticity and several combinations of finite elements. The mathematical formulations considered are (1) two-field formulation (displacement and fluid pressure), similar to our paper, (2) three-field formulation (displacement, velocity and fluid pressure) and (3) four-field formulation (displacement, solid pressure, velocity and fluid pressure). These different formulations were numerically investigated using several combinations of standard triangular and quadrilateral finite elements with different combinations of polynomial orders or continuities for the field variables. Lagrangian, Raviart-Thomas, Crouzeix-Raviart (triangular) and Rannacher-Turek (quadrilateral) elements were used for the numerical studies. For the low permeability problem, and as expected, equal order triangular and quadrilateral Lagrange finite elements (both first- and second-order) were found to fail in removing pressure oscillations. Mixed triangular and quadrilateral Lagrange elements (second-order for displacement and first-order for fluid pressure) were found to improve the pressure solution, but local pressure spikes were still exhibited. Satisfactory solutions were obtained only when using three- and four-field formulations with different element types for the field variables for example when representing the velocity field with linear Raviart-Thomas elements where the displacement and fluid pressure use second- and lowest-order triangular and quadrilateral Lagrangian elements. Our IGA analysis results produced satisfactory results based on the two-field formulation, without the need of introducing additional field variables and combinations of finite element types for the field variables. Figure 9: Numerical solution to the Haga problem using a graded mesh with small elements near the boundary layer and \(\Delta t=1s\). All figures are shown after two time steps. On the left the mixed order method, and on the right the equal order method. The continuity in the boundary layer is \(C\gamma^{-1}\) in the top row, and \(C^{0}\) in the bottom row.