Finite element methods in linear poroelasticity: theoretical and computational results
Linear Poroelasticity refers to fluid flow within a deformable porous medium under the assumption of relatively small deformations. Some of the areas that are being modeled with the equations of linear poroelasticity include reservoir engineering, soil mechanics and, more recently, biomedical engineering. The purpose of this dissertation is to present original results for the development, analysis and application of numerical finite element algorithms in the field of linear poroelasticity. A fully coupled finite element method involving continuous elements for displacements and a mixed space for flow is developed (CG/Mixed). Existence, uniqueness and optimality results are provided. The norm measuring the pressure error, however, depends on the value of the constrained specific storage coefficient. For degenerate values, this leads to a slightly weaker optimality result. For the not untypical case of a null constrained specific storage coef- ficient, the solution produced by the CG/Mixed scheme sometimes produces non-physical pressure oscillations, a phenomenon referred to as locking. One potential remedy is to eliminate the continuity requirement for the elements approximating displacements. Therefore, a family of schemes which couples discontinuous elements for displacements and a mixed space for flow is introduced (DG/Mixed). Existence and uniqueness are established, optimal a priori error estimates are provided, and some success in the removal of locking is shown. Direct verification of several benchmark analytical solutions shows that solutions in linear poroelasticity can lack regularity. This sometimes manifests in pressure boundary layers which might degrade the rate of convergence of numerical solutions. The situation can often be ameliorated with the development of adaptive grid refinement strategies. This motivates a posteriori estimates in terms of computable residual quantities. Interestingly, it is also shown that the CG/Mixed method can be combined with adaptive grid refinement as an alternative means to eliminate locking. The produced algorithms are then applied to some interesting application areas. In one instance, they are used to analyze the deformation and pressure dynamics in a cantilever bracket. Additionally, a variety of miscellaneous problems ranging from subsidence and well placement to scuba suit design highlight intriguing applications.