Theresa Köfler,
"Symbolic local Fourier Analysis to determine the inf-sup stability of the Stokes equations"
, Serie RISC Report Series, RISC, JKU, Hagenberg, Linz, 2023

Original Titel:

Symbolic local Fourier Analysis to determine the inf-sup stability of the Stokes equations

Sprache des Titels:

Englisch

Original Kurzfassung:

We are interested in a stable discretization of the Stokes equations, which arise in fluid mechanics and model the flow of fluids with slow velocities. For the analysis, we are focused on statements on the existence and the uniqueness of a solution. Moreover, we want to have information about the stable dependence of the solution on the data. Brezzi?s theorem guarantees us the existence of a unique solution that depends continuously on the input data, if certain conditions are fulfilled, which hold for the continuous form of the Stokes equations.
In order to solve the problem computationally, we have to discretize it, because we have to transfer the problem from an infinite number of degrees of freedom to finitely many ones. Again, we are interested in a result on existence and uniqueness of a solution and its dependence on the data, Moreover, we are focussed in discretization error estimates. Again, we apply Brezzi?s theorem. All conditions immediately carry over to the discrete form, but with one exception, namely the inf-sup condition. So, we want to focus on the constant, which is given by the inf-sup condition for the discrete inf-sup constant that is independent from the mesh size. In the literature, we can find some existing qualitative statements on the discrete inf-sup constant, but we want to get an explicit statement.
Such an explicit statement is obtained by local Fourier analysis. In order to get the inf-sup constant, we have to find the minimum of a rational function. We choose cylindrical algebraic decomposition (CAD) to calculate the minimum exactly. Alternatively, a minimum of a function could be derived exactly by curve sketching, but this is computationally too expensive in our case. Another alternative is to approximate the minimum by sampling. This method is fast but it does not deliver an exact solution. Therefore, it is no option for us. So, both of these alternatives are not relevant in our case and we stick to CAD to compute the minimum on a bounded domain. Nevertheless, the task of deriving the minimum with the help of CAD is not easy, because the computation gets more expensive with higher spline degree p. So, it is necessary to do some modifications like splitting the rational function into numerator and denominator, lowering the polynomial degree and using symmetry.
In this thesis, all computations are done in Mathematica and the results can be found on the webpage https://www3.risc.jku.at/people/vpillwei/koefler-slFA/.