Advancing the Mimetic Spectral Element Method

Abstract

Mimetic discretisation techniques are a growing field in computational physics research. Among these techniques, the recently developed mimetic spectral element method allows for exact discretisation of metric independent relations. This has been proven numerically in various mixed formulations, for instance the mixed velocity-vorticity-pressure formulation for Stokes flow, where mass conservation was point-wise strongly satisfied by the solution in the computational domain. Another example is the mixed stress-displacement formulation for the linear elasticity equations, where the balance law of linear momentum was point-wise strongly satisfied as well. A recent extension to a hybrid method leads to additional attractive features, such as the ability to decompose a large part of the computation of the solution into smaller problems. The aim of the research is to find a formulation for linear elasticity that is hybridisable while strongly satisfying conservation of linear and angular momentum as well, where the combination of linear momentum conservation and symmetry of the stress tensor is equivalent to angular momentum conservation. The proposed formulation has a mixed basis of both primal and algebraic dual nodal and edge basis functions. It fulfils the requirements as it is shown to be hybridisable, to satisfy point-wise linear momentum, and the discrete representation of the stress tensor is point-wise symmetric, hence angular momentum conservation is point-wise satisfied as well. The thesis furthermore functions as an overview of the method applied to elliptic problems, showing the results for previous formulations, and as a starting point for the next steps towards applying the method to fluids. A first step is proposed on extending the new formulation to a Stokes flow formulation with the stress as primary unknown, aimed at satisfying both linear and angular momentum conservation as well as mass conservation.