Least-squares PGD mimetic spectral element methods for systems of first-order PDEs

Sadeghi Namaghi, Layla K. 2025. Least-squares PGD mimetic spectral element methods for systems of first-order PDEs. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

This thesis is concerned with the efficient approximation of solutions to high-dimensional partial differential equations (PDEs) using the Proper Generalized Decomposition (PGD). The PGD represents unknown fields as a series of low-dimensional problems, which can overcome the computational challenge associated with the ‘curse of dimensionality’. How ever, convergence of PGD algorithms is not guaranteed, particularly when using Galerkin formulations. To address these limitations, PGD algorithms based on rigorously derived Least-Squares formulations are developed based on optimal first-order reformulations of the problems. The theoretical framework is based on Agmon-Douglis-Nirenberg (ADN) theory, which is used to address practical issues and convergence considerations. A novel approach to solving elliptic PDEs is presented based on a PGD which solves the minimization of a discrete rather than a continuous Least-Squares functional. The spectral element method which combines the flexibility of finite element methods with the accuracy of spectral methods is used as the foundation of the discretization. The application of the Least-Squares PGD algorithm to the Poisson, convection-diffusion and the Stokes problems demonstrates the efficiency, robustness and precision of this novel approach. Notably, the algorithm maintains high accuracy for convection-dominated prob lems, which are prone to numerical instability. The optimal Least-Squares PGD is extended to incorporate a mimetic framework for solving the Poisson problem. This ensures that the algebraic equations obtained through discretization are compatible and physically consistent, providing accurate numerical rep resentations of the physical solution. Results validate, for the first time, the successful implementation of the mimetic approach within a Least-Squares PGD algorithm. The per formance and properties of the Least-Squares PGD algorithm on the series of benchmark problems studied in this thesis demonstrate the efficiency and accuracy of this approach for solving PDEs and provide a solid foundation for further development of this technique.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Uncontrolled Keywords:	1. Least-Squares 2. ADN theory 3. Proper Generalized Decomposition 4. Mimetic methods 5. Spectral Element Method
Funders:	EPSRC
Date of First Compliant Deposit:	19 May 2026
Last Modified:	19 May 2026 15:02
URI:	https://orca.cardiff.ac.uk/id/eprint/187112

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