Commit 69b38b83 by Tilman Aleman

### 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, forc=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 withc=1$are defined by the$L_2\$-orthogonal (but not normalized) basis \begin{align*} ... ...
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