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University of Texas
===firstname:
Bryan, D
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===lastname2:
Biros
===lastname:
Quaife
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===email:
quaife@ices.utexas.edu
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===postal:
ICES
University of Texas
201 East 24th Street
Stop C0200
Austin, Texas
78712-1229
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===ABSTRACT:
\newcommand{\xx}{{\mathbf{x}}}
\newcommand{\yy}{{\mathbf{y}}}
\newcommand{\ff}{{\mathbf{f}}}
\renewcommand{\SS}{{\mathcal{S}}}
\newcommand{\RR}{{\mathbb{R}}}
\newcommand{\BB}{{\mathcal{B}}}
\newcommand{\DD}{{\mathcal{D}}}
\newcommand{\TT}{{\mathcal{T}}}
\newcommand{\LL}{{\mathcal{L}}}

We are interested in simulating high concentration suspensions of deformable capsules suspended in a Stokesian fluid. Such capsules are typically modeled as infinitesimally thin membranes filled with a viscous fluid.  In particular, we are interested in a specific type of capsule known as a {\em vesicle}.  Vesicles are closed inextensible lipid membranes containing a viscous fluid.  The membrane mechanics are characterized by bending resistance (derived by a Helfrich energy, or a minimization of the $L^2$ norm of the mean curvature), zero resistance to shear, and inextensibility (infinite resistance to elongation).  Although we focus on vesicles, the resulting methodologies should be applicable in a broad range of phenomena in complex fluids and suspensions.

Vesicle evolution dynamics are given by a quasi-static Stokes equation driven by interface jump conditions derived by a balance of forces on the membrane.  Given the position of the vesicles, the jump conditions can be evaluated, a Stokes problem with interface conditions can be solved, and the velocity at the interface advances the vesicle to its new location.  We use a boundary integral formulation that results in a system of non-linear integro-differential equations for the vesicles's position and an auxiliary field that is a ``tension'' (it acts as a Lagrange multiplier to enforce the inextensibility constraint). Below we summarize the main equations.

Consider a single vesicle $\gamma \in \RR^{2}$ parameterized by the periodic function $\xx$.  The equations governing the motion of the vesicle are
\begin{align*}
  \frac{d\xx}{dt} = \SS[-\xx_{ssss}+(\sigma \xx_{s})_{s}], 
  \hspace{20pt} \xx_{s} \cdot \frac{d\xx_{s}}{dt} = 0,
\end{align*}
where $\sigma$ is the tension and $\SS[\ff](\xx) = \int_{\gamma}G(\xx-\yy)\ff(\yy)ds_{\yy}$, where $G(\xx) = \frac{1}{4\pi\mu}\left(-\log|\xx| + \frac{\xx \otimes \xx}{ \xx|^{2}}\right)$, is the single-layer potential for Stokes flow.  We introduce the operators $\BB$ (bending), $\TT$ (tension), and $\DD$ (divergence) so that the governing equations are 
\begin{align*}
  \frac{d\xx}{dt} = \SS[-\BB\xx+\TT\sigma], \hspace{20pt}
  \DD\frac{d\xx}{dt} = 0.
\end{align*}

Discretizing in time and linearizing, the vesicle position and tension are updated by solving
\begin{align}
  \left[ \begin{array}{cc}
  I + \Delta t \SS\BB & -\Delta t \SS\TT \\
  -\DD\SS\BB & \DD\SS\TT 
  \end{array} \right]
  \left[ \begin{array}{c}
  \xx^{N+1} \\ \sigma^{N+1}
  \end{array} \right] = 
  \left[ \begin{array}{c}
  \xx^{N} \\ 0
  \end{array} \right]. 
  \label{e:update}
\end{align}
This system is solved with a matrix-free fast-multipole accelerated GMRES for the new position and tension. However, the number of iterations depends on $N$ since the system is ill-conditioned. This motivates the use of multigrid. In this talk we present the overall scheme and preliminary results that demonstrate the effectiveness of the proposed scheme.

As a first step, we are looking for efficient solvers for the Schur complement of the tension, $\LL:=\DD\SS\TT$.  $\LL$ must be inverted for explicit schemes or it can be used to form block preconditioners for implicit schemes when solving~\eqref{e:update}.  Standard smoothers applied to integral operators such as $\LL$ generally reduce low frequencies in the error.  We propose using GMRES applied to the high frequency components as a smoother.  In more detail, the smoother for $\LL \sigma = b$ is $\sigma^{N+1} = P \sigma^{N} + \sigma_{H}$, where $\sigma_{H}$ solves $(I-P) \LL \sigma_{H} = (I-P) b - (I-P)\LL P\sigma^{N}$, and $P$ is the projection onto the low frequency space.

We use a V-cycle multigrid solver for $\LL$ and report results in Table~\ref{t:multiSolver}.  We see that only a few GMRES iterations are required to apply the smoother.  We set the GMRES tolerance of the smoother to $10^{-8}$, the coarsest level to $N=8$, and stop the solver when the relative error is less than $10^{-8}$.  We are currently experimenting with using multigrid to precondition a GMRES solver for $\LL$.

\begin{table}[h]
\centering
\begin{tabular}{cccc} 
N & $\rho$ & $nV$ & GMRES \\ 
\hline 16 & 0.138 & 10 & 2 \\ 
32 & 0.365 & 19 & 4 \\ 
64 & 0.448 & 23 & 8 \\ 
128 & 0.573 & 35 & 13 \\
\end{tabular} 
\caption{\label{t:multiSolver} $N$ is the problem size, $\rho$ is the average convergence rate, $nV$ is the number of V-cycles, and GMRES is the number of GMRES iterations required by the smoother at the finest level.} 
\end{table}
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===title:
Multigrid Applied to Two-Dimensional Vesicle Suspensions
===firstname2:
George