Title: An Embedded Boundary Integral Solver for the Stokes Equations 

(NYU-CS-TR837)

Authors: George Biros, Lexing Ying, and Denis Zorin 

Abstract:
We present a new method for the solution of the Stokes equations. Our
goal is to develop a robust and scalable methodology for two and three
dimensional, moving-boundary, flow simulations. Our method is based on
Anita Mayo's method for the Poisson's equation: ``The Fast
Solution of Poisson's and the Biharmonic Equations on Irregular
Regions'', SIAM J. Num. Anal., 21 (1984), pp. 285--299. We embed the
domain in a rectangular domain, for which fast solvers are available,
and we impose the boundary conditions as interface (jump) conditions
on the velocities and tractions. We use an indirect boundary integral
formulation for the homogeneous Stokes equations to compute the
jumps. The resulting integral equations are discretized by
Nystrom's method.  The rectangular domain problem is discretized
by finite elements for a velocity-pressure formulation with equal
order interpolation bilinear elements (Q1-Q1). Stabilization is
used to circumvent the inf-sup condition for the pressure
space. For the integral equations, fast matrix vector multiplications
are achieved via a N log N algorithm based on a block representation
of the discrete integral operator, combined with (kernel independent)
singular value decomposition to sparsify low-rank blocks.  Our code is
built on top of PETSc, an MPI based parallel linear algebra
library. The regular grid solver is a Krylov method (Conjugate
Residuals) combined with an optimal two-level
Schwartz-preconditioner. For the integral equation we use GMRES. We
have tested our algorithm on several numerical examples and we have
observed optimal convergence rates.