Commit 69b38b83 authored by Tilman Aleman's avatar Tilman Aleman
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expanding doc

parent 2694efca
......@@ -19,7 +19,7 @@
\newcommand{\fut}{\mathbf{u}_\tau}
\newcommand{\fvt}{\mathbf{v}_\tau}
\newcommand{\blinner}[2]{(#1,#2)_{L^2(\Gamma)}}
\newcommand{\normal}[0]{\mathbf{n}}
\newcommand{\bldinner}[2]{(#1,#2)_{L^2(\Gamma_h)}}
\newcommand{\taylorhood}[1]{\mathbfcal{P}_{#1}-\mathcal{P}_{#1-1}}
\newcommand{\dsgrad}{\nabla_{\Gamma_h}}
......@@ -36,19 +36,18 @@
% START HERE %
% ------------------------------------------ %
\title{A trace finite element method for the surface Stokes equations} % Replace with appropriate title
\title{A trace finite element method for the surface Vector-Laplacian} % Replace with appropriate title
% \author{Author's Name\\Math 790, Real Analysis} % Replace "Author's Name" with your name
\date{\vspace{-5ex}}
\maketitle
\section{Continuous Problem}
In the following, we will mostly use the notation and results given in \cite{TJ_PHD_2021}.
We consider the surface Stokes problem, which for a given (sufficiently regular) surface $\Gamma\subset \R^3$ reads as follows:
We consider the surface Vector-Laplacian, which for a given (sufficiently regular) surface $\Gamma\subset \R^3$ reads as follows:
{\begin{problem}
For a given tangential force vector field $\mathbf{f}_\tau: \Gamma \rightarrow \R^3$ and a source term $g: \Gamma \rightarrow \R$, determine a tangential velocity $\mathbf{u}_\tau : \Gamma \rightarrow \R^3$ with $\fut \perp \mathcal{K}:=\ker E$ and surface pressure $p:\Gamma \rightarrow \R$ such that
For a given tangential force vector field $\mathbf{f}_\tau: \Gamma \rightarrow \R^3$, determine a tangential velocity $\mathbf{u}_\tau : \Gamma \rightarrow \R^3$ with $\fut \perp \mathcal{K}:=\ker E$ such that
\begin{align*}
-\mathbf{P} \surfdiv (E(\mathbf{u}_\tau)) +\sgrad p &= \mathbf{f}_\tau & \text{on } \Gamma,\\
\surfdiv \fut &= g &\text{on } \Gamma.
-\mathbf{P} \surfdiv (E(\mathbf{u}_\tau)) +\sgrad p &= \mathbf{f}_\tau & \text{on } \Gamma
\end{align*}
Here, $E(\fut)$ is the \emph{surface rate-of-strain tensor}, defined by
\begin{align*}
......@@ -57,25 +56,27 @@ Here, $E(\fut)$ is the \emph{surface rate-of-strain tensor}, defined by
and $\mathbf{P}=\mathbf{I}-\mathbf{n}\mathbf{n}^T$ is the tangential projection, where $\mathbf{n}$ is the outer normal.
\end{problem}}Defining
\begin{align*}
a(\mathbf{u},\mathbf{v})&:= \int_\gamma E(\mathbf{u}):E(\mathbf{v}) ds,\\
b(\mathbf{u},p)& := -\int_\Gamma p\surfdiv\mathbf{u} ds,
a(\mathbf{u},\mathbf{v})&:= \int_\Gamma E(\mathbf{u}):E(\mathbf{v}) ds, \\
k(u,v) &:= \eta \int_\Gamma (\mathbf{u}\cdot \normal)(\mathbf{v}\cdot \normal) ds
\end{align*}
and
\begin{align*}
\mathbf{V}&:=\mathbf{H}^1(\Gamma),\\
\mathbf{V}_\tau &:= \set{\mathbf{u}\in \mathbf{V}\vert \mathbf{u}\cdot \mathbf{n}=0}
\mathbf{V}_\tau &:= \set{\mathbf{u}\in \mathbf{V}\vert \mathbf{u}\cdot \mathbf{n}=0}, \\
\mathbf{V}_* &:= \set{\mathbf{u}\in \mathbf{L}^2(\Gamma)\vert \ \fut \in \mathbf{V}_\tau, u_N \in L^2(\Gamma)}
\end{align*}
the corresponding variational formulation can be written as
\begin{problem}\label{continuous}
For a given force field $\mathbf{f}\in \mathbf{L}^2(\Gamma)$ with $\mathbf{f}\cdot \mathbf{n}=0$ and a source term $g \in L^2_0(\Gamma)$ determine $(\fut, p) \in \mathbf{V}_\tau \times L^2_0(\Gamma)$ with $\fut \perp \mathcal{K}$ such that
For a given force field $\mathbf{f}\in \mathbf{L}^2(\Gamma)$ with $\mathbf{f}\cdot \mathbf{n}=0$, determine $\fut \in \mathbf{V}_\tau$ with $\fut \perp \mathcal{K}$ such that
\begin{align*}
a(\fut, \fvt)+b(\fvt, p) &= \blinner{\mathbf{f}}{\fvt} & \forall \fvt \in \mathbf{V}_\tau,\\
b(\fut, q) &= -\blinner{g}{q} & \forall q \in L^2(\Gamma)
a(\fut, \fvt)&= \blinner{\mathbf{f}}{\fvt} & \forall \fvt \in \mathcal{K}^\perp
\end{align*}
\end{problem}
\subsection{Stability}
The problem is well-posed, this follows directly from the well-posedness of \cite[Continuous Problem 5.10]{TJ_PHD_2021}
\section{Discrete formulation}
To approximate \cref{continuous}, we consider a $\taylorhood{k}$ trace Taylor-Hood finite element discretization based on the higher order parametric trace finite element spaces using a consistent penalty approach. For details regarding the finite element spaces, see \cite[Section 3.2.1]{TJ_PHD_2021}.
To approximate \cref{continuous}, we consider the higher order parametric trace finite element spaces using a consistent penalty approach. For details regarding the finite element spaces, see \cite[Section 3.2.1]{TJ_PHD_2021}.
Defining the discrete rate-of-strain tensor
\begin{align*}
......@@ -83,8 +84,7 @@ Defining the discrete rate-of-strain tensor
\end{align*}
with $\Gamma_h$ being the discrete approximation of the surface $\Gamma$, $\dsgrad:= \mathbf{P}_h \nabla $, $\mathbf{P}_h$ the tangential projection on $\Gamma_h$, $\mathbf{n}_h$ the outer normal on $\Gamma_h$ and $\mathbf{H}_h:=\nabla(\Pi_{\Theta}^{k-1}(\mathbf{n_h}))$ an approximation to the Weingarten map allows us to define
\begin{align*}
a_{\tau,h}&:=\ghint{E_{\tau,h}(\mathbf{u}):E_{\tau,h}(\mathbf{v})} ,\\%+ \epsilon \ghint{\mathbf{P}_h \mathbf{u}\cdot \mathbf{P}_h\mathbf{v}},\\
b_h(\mathbf{u},q)&:=\ghint{\mathbf{u}\cdot \dsgrad q\ }.
a_{\tau,h}(\mathbf{u},\mathbf{v})&:=\ghint{E_{\tau,h}(\mathbf{u}):E_{\tau,h}(\mathbf{v})}%+ \epsilon \ghint{\mathbf{P}_h \mathbf{u}\cdot \mathbf{P}_h\mathbf{v}},\\
\end{align*}
Further, as our function space is defined on the volume but we are only interested in tangential solutions, we define the penalty form
\begin{align*}
......@@ -92,15 +92,14 @@ Further, as our function space is defined on the volume but we are only interest
\end{align*}
and to achieve stability, we use normal gradient volume stabilization for both the velocity and pressure variables:
\begin{align*}
s_h(\mathbf{u},\mathbf{v})&:=\rho_u \int_{\Omega^\Gamma_{\Theta}}(\nabla \mathbf{u}\mathbf{n}_h)\cdot (\nabla \mathbf{u}\mathbf{n}_h) dx,\\
\tilde{s}_h(p,q)&:=\rho_p \int_{\Omega^\Gamma_{\Theta}}(\mathbf{n}_h\cdot \nabla p)(\mathbf{n}_h\cdot \nabla q) dx.
s_h(\mathbf{u},\mathbf{v})&:=\rho_u \int_{\Omega^\Gamma_{\Theta}}(\nabla \mathbf{u}\mathbf{n}_h)\cdot (\nabla \mathbf{u}\mathbf{n}_h) dx
%\tilde{s}_h(p,q)&:=\rho_p \int_{\Omega^\Gamma_{\Theta}}(\mathbf{n}_h\cdot \nabla p)(\mathbf{n}_h\cdot \nabla q) dx.
\end{align*}
Now, we can state the discrete surface Stokes problem:
\begin{problem}
Find $(\mathbf{u}_h, p_h) \in \mathbf{U}_h \times Q_h$ such that
Find $\mathbf{u}_h\in \mathbf{U}_h $ such that
\begin{align*}
A_{\tau,h}(\mathbf{u}_h.\mathbf{v}_h)+ b_h(\mathbf{v}_h.p_h) &= \bldinner{\mathbf{f}_h}{\mathbf{v}_h}&\forall \mathbf{v}_h \in \mathbf{U}_h, \\
b_h(\mathbf{u}_h, q_h)-\tilde{s}_h(p_h, q_h) &= \bldinner{-g_h}{q_h} & \forall q_h \in Q_h
A_{\tau,h}(\mathbf{u}_h.\mathbf{v}_h)&= \bldinner{\mathbf{f}_h}{\mathbf{v}_h}&\forall \mathbf{v}_h \in \mathbf{U}_h
\end{align*}
with $A_{\tau,h}\dotargs :=a_{\tau,h}\dotargs + s_h\dotargs + k_h\dotargs$
\end{problem}
......@@ -113,8 +112,7 @@ We consider the surface vector laplace equation on surfaces $\Gamma_c \subset \O
\end{align*}
Obviously, for $c=1$ this describes the unit sphere and for $c>1$ we get ellipsoids. Additionally, with $\mathbf{P}_{\mathcal{K}^\perp}$ being the projeciton on the orthogonal complement of $\mathcal{K}$, we prescribe the solution
\begin{align*}
\mathbf{u}(\mathbf{x})&:=\mathbf{P}_{\mathcal{K}^\perp}\left(\mathbf{P}\left(-z^2,x,y\right)^T\right), \\
p(\mathbf{x}) &:= xy^2+z.
\mathbf{u}(\mathbf{x})&:=\mathbf{P}_{\mathcal{K}^\perp}\left(\mathbf{P}\left(-z^2,x,y\right)^T\right)
\end{align*}
The Killing vector fields on this geometry with $c=1$ are defined by the $L_2$-orthogonal (but not normalized) basis
\begin{align*}
......
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