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

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Many mathematical models that describe problems in science and engineering are defined in high dimensional spaces. Examples can be found in a diverse range of application areas such as quantum chemistry, kinetic theory descriptions of materials (including complex fluids), the chemical master equation governing many biological processes (e.g. cell signalling) and models of financial mathematics (e.g. option pricing). The mathematical description of these problems is invariably in terms of a system of partial differential equations (PDEs). For practical problems these systems do not possess analytical solutions and therefore it is necessary to solve them numerically. It is important that the system of algebraic equations obtained as a result of discretisation is a compatible (mimetic) and physically consistent system so that the numerical approximation is an accurate representation of the physical solution to the problem. The governing systems of PDEs is written in terms of an equivalent system of first-order differential equations which is subsequently formulated in terms of a least squares functional. Effectively the solution of a system of PDEs is converted into an unconstrained minimisation problem. The exceptional stability of least-squares formulations has led to the widespread use of low-order finite elements in their discretization. Unfortunately, these 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 is crucially important.