Partition of Unity Finite Element Method for Solving Phase Change Problems

A Master of Science in Mathematics by Abdulla Hussain Kayyani entitled, “Partition of Unity Finite Element Method for Solving Phase Change Problems”, submitted in November 2023. Thesis advisor is Dr. Youssef Belhamadia. Soft copy is available (Thesis, Completion Certificate, Approval Signatures, and AUS Archives Consent Form).

Abstract

This thesis presents a comprehensive study of the Partition of Unity Finite Element Method (PUFEM) and its applications in solving phase change problems. First, it introduces the Finite Element Method (FEM), the classical counterpart to PUFEM, and a pivotal tool in simulating phenomena described by PDES. The limitations of FEM when handling problems with steep gradients are highlights. The thesis is dedicated to the exploration and development of PUFEM, and how it is considered as an advanced variant of FEM that incorporates enrichment functions to enable more accurate numerical solutions. The development of PUFEM is presented for the Poisson Equation, Heat Equation, and Bistable Equation showing its superior accuracy and efficiency compared to the standard FEM. Finally, the application of PUFEM for phase change problems is illustrated with detailed numerical results showing its superiority to the standard FEM offering avenues for future research.