Multigrid Methods for the Indefinite Helmholtz Equation

Student Number

05299

Degree

Master of Science in Mathematics

Department

Department of Mathematical Sciences

Faculty/School

School of Mathematics and Computer Science (SMCS)

Date of Award

Fall 2025

Advisor

Dr. Hisham bin Zubair Syed, Assistant Professor, Department of Mathematical Sciences, School of Mathematics and Computer Science (SMCS)

Committee Member 1

Dr. Abdul Majid, Examiner, Assistant Professor & Program Coordinator SMCS, IBA

Committee Member 2

Dr. Amir Bashir, Assistant Professor and Chairperson of Mathematical Sciences, Department of Mathematical Sciences

Committee Member 3

Dr. Shakeel Ahmed Kamboh, Reviewer, Associate Professor, QUEST

Project Type

Thesis

Access Type

Restricted Access

Document Version

Final

Pages

xii, 70

Keywords

Indefinite Helmholtz Equation, Multigrid Methods, Exterior Complex Scaling, Cell-centered grid discretization, Complex Shifted Laplacian Preconditioner

Abstract

The indefinite Helmholtz equation is a notoriously challenging problem in numerical analysis, particularly due to its oscillatory nature and indefiniteness at high wavenumbers. Classical iterative solvers and standard multigrid methods often struggle to converge efficiently, especially in heterogeneous media. This thesis investigates the underlying numerical difficulties associated with the Helmholtz equation and explores specialized multigrid strategies designed to address its complexities. At the core of this work is the development of a multigrid-based Complex Shifted Laplacian (CSL) preconditioner, discretized on a cell-centered grid and enhanced with higher-order inter-grid transfer operators, ILU(0) smoothing, and Exterior Complex Scaling (ECS) for boundary treatment. Numerical experiments on both constant and spatially varying wavenumber models demonstrate the robustness and scalability of the proposed approach.

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