Bryan, D Quaife
Multigrid Applied to Two-Dimensional Vesicle Suspensions

ICES
University of Texas
201 East 24th St
Stop C0200
Austin
Texas 78712-1229
quaife@ices.utexas.edu
George Biros

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 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 $\gamma \in {\mathbb{R}}^{2}$ parameterized by the periodic function ${\mathbf{x}}$ . The equations governing the motion of the vesicle are

$$\frac{d{\mathbf{x}}}{dt} = {\mathcal{S}}[-{\mathbf{x}}_{ssss}+(\s... ...})_{s}], \hspace{20pt} {\mathbf{x}}_{s} \cdot \frac{d{\mathbf{x}}_{s}}{dt} = 0,$$

where $\sigma$ is the tension and ${\mathcal{S}}[{\mathbf{f}}]({\mathbf{x}}) = \int_{\gamma}G({\mathbf{x}}-{\mathbf{y}}){\mathbf{f}}({\mathbf{y}})ds_{{\mathbf{y}}}$ , where $G({\mathbf{x}}) = \frac{1}{4\pi\mu}\left(-\log\vert{\mathbf{x}}\vert + \frac{{\mathbf{x}}\otimes {\mathbf{x}}}{\vert{\mathbf{x}}\vert^{2}}\right)$ , is the single-layer potential for Stokes flow. We introduce the operators ${\mathcal{B}}$ (bending), ${\mathcal{T}}$ (tension), and ${\mathcal{D}}$ (divergence) so that the governing equations are

$$\frac{d{\mathbf{x}}}{dt} = {\mathcal{S}}[-{\mathcal{B}}{\mathbf{x}}+{\mathcal{T}}\sigma], \hspace{20pt} {\mathcal{D}}\frac{d{\mathbf{x}}}{dt} = 0.$$

Discretizing in time and linearizing, the vesicle position and tension are updated by solving

$$\left[ \begin{array}{cc} I + \Delta t {\mathcal{S}}{\mathcal{B}}&... ...y} \right] = \left[ \begin{array}{c} {\mathbf{x}}^{N} 0 \end{array} \right].$$ (1)

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, ${\mathcal{L}}:={\mathcal{D}}{\mathcal{S}}{\mathcal{T}}$ . ${\mathcal{L}}$ must be inverted for explicit schemes or it can be used to form block preconditioners for implicit schemes when solving (). Standard smoothers applied to integral operators such as ${\mathcal{L}}$ 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 ${\mathcal{L}} \sigma = b$ is $\sigma^{N+1} = P \sigma^{N} + \sigma_{H}$ , where $\sigma_{H}$ solves $(I-P) {\mathcal{L}}\sigma_{H} = (I-P) b - (I-P){\mathcal{L}}P\sigma^{N}$ , and $P$ is the projection onto the low frequency space.

We use a V-cycle multigrid solver for ${\mathcal{L}}$ and report results in Table . 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 ${\mathcal{L}}$ .

Table: $N$ 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.

N	$\rho$	$nV$	GMRES
16	0.138	10	2
32	0.365	19	4
64	0.448	23	8
128	0.573	35	13