Analysis and finite element approximation for nonlinear problems in poroelasticity and bioconvection

Abstract

This dissertation is concerned with nonlinear systems of partial differential equation with solution dependent physical coefficients satisfying the Nemytskii assumption. Such equations arise from two important application fields: poroelasticity and bioconvection.

First, we consider a quasi-static poroelasticity model with dilatation dependent hydraulic conductivity and an implicit time derivative. We derive the existence and uniqueness of solutions using the modified Rothe's method, Brouwer's fixed point Theorem and the Sobolev embedding Theorem. Next we construct a finite element approximation with linear elements and establish the optimal error estimate. We then conduct numerical examples to verify the convergence and simulate the diffusion in a fluid saturated sponge.

Second, we study the bioconvection model, a coupled Navier-Stokes type equation, with concentration dependent viscosity. We combine the theory of the Navier-Stokes equation and the modified Rothe's method to establish existence and uniqueness of solutions of both steady and time dependent bioconvection. After that we perform finite element analysis with Taylor-Hood elements and prove the convergence theorem. Finally numerical examples are constructed using lab data to verify the convergence of the numerical scheme and simulate the convection pattern formed by micro-organisms inside a culture fluid.