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Least Squares PGD Mimetic Spectral Element Methods for Systems of First-Order PDEs

Please note that this project is self-funded.

The goal of the project will be to develop a least squares PGD mimetic spectral element method for the Stokes problem. There will be theoretical and computational aspects to this strand of the research.

Least-squares spectral element methods for PDEs reformulate them into unconstrained minimisation problems. The exceptional stability of least-squares formulations has led to the widespread use of low-order finite elements in their discretization. Unfortunately, resulting finite element methods are only approximately conservative, which generally leads to violation of fundamental physical properties, such as loss of mass conservation. In many cases this drawback can outweigh the potential advantages of least squares methods. As a result, improving the conservation properties of least-squares methods has attracted significant attention.

Project aims and methods

The project will begin with a literature survey on least squares formulations of systems of first-order partial differential equations (PDEs), spectral element methods and mimetic spectral element methods. Alongside this, computational work involving the implementation of the standard spectral element method for standard Poisson problems in 1D and 2D will be performed using MATLAB.

The theoretical aspect of the project is concerned with an investigation of the convergence properties of the least squares mimetic spectral element method. Convergence proofs that exist will be analysed and used as a basis for extension to the Proper Generalized Decomposition (PGD) and rates of convergence of the PGD approximation will be investigated for the Poisson equation, in the first instance. A least squares mimetic spectral element method for the Poisson equation based on PGD will be developed. The Proper Generalized Decomposition (PGD) is an approach for developing efficient approximations for high-dimensional problems. Its use in conjunction with mimetic spectral or finite elements has never been investigated.

You will learn about different methods for discretising partial differential equations and iterative methods for solving the resulting systems of algebraic equations. You will also develop an understanding about the theoretical properties of partial differential equations. There will also be an element of scientific computing and the student will develop programming skills using MATLAB, C++ etc.


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Yr Athro Tim Phillips

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