Stability analysis for linearized Euler

The unsteady linearized Euler problem

For a given velocity field \(\mathbf{w}\) find \(\mathbf{u},p\) so that \begin{align} \partial_t \mathbf{u} + \operatorname{div}( \mathbf{u} \otimes \mathbf{w} ) - \nabla p & = f, \\ \operatorname{div}( \mathbf{u} ) & = 0, \end{align} complemented by appropriate initial and boundary conditions. The aim of the thesis is to understand stability criteria for finite element methods w.r.t. the treatment of the transport term \(\operatorname{div}( \mathbf{u} \times \mathbf{w} )\). More specifically we are interested in the investigation of an \(H(\operatorname{div})\)-conforming Upwind discretization, similar to the one considered in [1].

A related eigenvalue problem

As one approach to investigate stability, we are interested in the solution to the following eigenvalue problem: For a given velocity field \(\mathbf{w}\) find \(\omega,\mathbf{u},p\) so that \begin{align} i \omega \mathbf{u} + \operatorname{div}( \mathbf{u} \otimes \mathbf{w} ) - \nabla p & = f, \\ \operatorname{div}( \mathbf{u} ) & = 0, \end{align} complemented by appropriate boundary conditions.

We are interested in the connection between the stability of differente FEM discretizations and the approximation of the eigenvalue problems.

Steps:

1. Review the stability analysis of \(H(\operatorname{div})\)-conforming Upwind discretization, especially [1].
2. Set up the eigenvalue solver for a generic FEM discretization.
3. Evaluate different FEM discretization w.r.t. the eigenvalue problem and the unsteady problem.
4. Ultimate goal: Optimal order error analysis for the \(H(\operatorname{div})\)-conforming Upwind discretization

References

[1] Gabriel Barrenechea, Erik Burman and Johnny Guzmán. Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow, Mathematical Models and Methods in Applied Sciences, 30(05):847-865, 2020.