# Finite element methods for surface vector partial differential equations

Jankuhn, Thomas; Reusken, Arnold (Thesis advisor); Olshanskii, Maxim (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

In this thesis we develop and analyze efficient and highly accurate finite element methods for the numerical simulation of the surface vector-Laplace equation and the surface (Navier-)Stokes equations on a stationary surface. First, the incompressible surface Navier-Stokes equations on an evolving surface are derived. The derivation is based on fundamental continuum mechanical principles for a viscous material surface embedded in an ambient continuum medium. The resulting equations are formulated in terms of tangential differential operators in Cartesian coordinates, which makes the formulation more convenient for our numerical purposes. We use a directional splitting of the system into a coupled system of equations for the tangential density flow and the normal velocity of the surface in order to gain a further insight into both of these components. On an a priori given stationary surface the normal velocity vanishes and one is interested in the tangential density flow only. One can deduce the following three simplified models on a stationary surface: the surface vector-Laplace equation, the surface Stokes equations and the surface Navier-Stokes equations. In surface flow problems we have the constraint that the flow must be tangential to the surface. It is not obvious how this constraint should be treated numerically. To better understand this issue we first consider discretizations of the surface vector-Laplace equation. We use the higher order parametric trace finite element method. Three different natural techniques for treating the tangential constraint are studied, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. A complete analysis of all three methods is presented that reveals how the discretization error bounds in the energy norm depend on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the approximation of the surface, the penalty parameter, the order of the normal vector approximation used in the penalty terms in both penalty methods and the degree of polynomials used for the approximation of the Lagrange multiplier. Furthermore, for the consistent penalty method we derive a new optimal $L^2$-error bound. The results are confirmed and illustrated by numerical experiments. In a comparison of the three methods we conclude that the consistent penalty method has significant advantages over the other two methods. For the surface Stokes equations an isoparametric $\boldsymbol{\mathcal{P}}_k$-$\mathcal{P}_{k-1}$ trace Taylor-Hood finite element discretization ($k\geq2$) is introduced. For the tangential constraint we employ the same consistent penalty technique used and analyzed for the surface vector-Laplace equation. With the help of a uniform inf-sup stability result for the Taylor-Hood pair well-posedness is shown and an optimal energy norm error bound is derived. Furthermore, we present a new error analysis that yields an optimal $L^2$-error bound. Results of numerical experiments confirm the error analysis. To discretize the surface Navier-Stokes equations on a stationary surface we apply an isoparametric $\boldsymbol{\mathcal{P}}_k$-$\mathcal{P}_{k-1}$ trace Taylor-Hood finite element method ($k\geq2$) in space and a second order accurate semi-implicit BDF$2$ method in time. For the case that no outer forces are present, asymptotic properties of the solution in the continuous case are derived and used as a validation of numerical simulation results.