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In this thesis we develop a stabilised finite element method for solving theequations of poroelasticity to enable solving complex models of biologicaltissues such as the human lungs. For the proposed numerical scheme, we use thelowest possible approximation order: piecewise constant approximation for thepressure, and piecewise linear continuous elements for the displacements andfluid flux. Due to the discontinuous pressure approximation, sharp pressuregradients due to changes in material coefficients or boundary layer solutionscan be captured reliably. We begin by developing theoretical results forapproximating the linear poroelastic equations valid in small deformations. Inparticular, we prove existence and uniqueness, an energy estimate and anoptimal a-priori error estimate for the discretised problem. We then extendthis work and construct a stabilised finite element method to solve theporoelastic equations valid in large deformations. We present the linearisationand discretisation for this nonlinear problem, and give a detailed account ofthe implementation. We rigorously test both the linear and nonlinear finiteelement method using numerous test problems to verify theoretical stability andconvergence results, and the method's ability to reliably capture steeppressure gradients. Finally, we derive a poroelastic model for lung parenchymacoupled to an airway fluid network model, and develop a stable method to solvethe coupled model. Numerical simulations, on a realistic lung geometry,illustrate the coupling between the poroelastic medium and the network flowmodel, and simulations of tidal breathing are shown to reproduce globalphysiologically realistic measurements. We also investigate the effect ofairway constriction and tissue weakening on the ventilation, tissue stress andalveolar pressure distribution.

A Low Order Finite Element Method for Poroelasticity with Applications to Lung Modelling