Date of Award

2025-05-01

Degree Name

Master of Science

Department

Mathematical Sciences

Advisor(s)

Yi Song-Young

Abstract

Linear poroelasticity theory describes the interaction between the motion of fluids and deformation of porous media. The theory serves various applications in a wide range of science and engineering fields, such as soil mechanics, oil reservoir modeling, and bio-medical applications. Partial differential equations are used to model the complex interaction between fluid flow and solid deformation in porous media. Since analytical solutions to these equations are rarely available for realistic problems, we resort to numerical solutions. One major advantage is their ability to provide accurate approximations of the solutions. Various numerical methods have been proposed to solve Biot's poroelasticity system. These methods are mainly classified into three main classes: monolithic methods, sequential methods, and iterative methods. The sequential methods decouple the system into flow and mechanics subproblems, then solve them sequentially one after the other at each time step. In this thesis we propose two sequential finite element methods to solve Biot's poroelasticity system. The methods both use stabilization terms to ensure convergence and stability of the displacement and pressure solutions, and differ mainly by the type of stabilization used. Particularly, the first uses a L2 -type stabilization term while the second uses an H 1 -type stabilization term. An extensive numerical convergence study of both sequential methods was conducted. First, we used two manufactured solutions with pure Dirichlet boundary conditions. The performance of both sequential methods is analyzed for two different parameter regimes. The H1 -type stabilization method proves more robust and delivers better and optimal convergence rates for both parameter regimes. Another numerical convergence study for the sequential method with H 1 -type stabilizing approach is carried out with another manufactured solution and mixed boundary conditions, where the approach offers optimal convergence rates. The performance of this method was also tested against the well-known Barry and Mercer problem.

Language

en

Provenance

Received from ProQuest

File Size

59 p.

File Format

application/pdf

Rights Holder

Jeff Frimpong Amponsah

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Included in

Mathematics Commons

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