An embedded boundary integral solver for the stokes equations

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, o w simulations. Our method is based on Anita Mayo's method for the Poisson's equation: iThe Fast Solution of Poisson's and the Biharmonic Equations on Irregular Regionsi, SIAM J. Num. Anal., 21 (1984), pp. 285n 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 Nystr¤ om's method. The rectangular domain problem is discretized by nite 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. In this paper we propose a boundary integral method for the steady two-dimensional Stokes equations in irregular domains. We have developed this method as a building block for a Navier-Stokes solver for problems with moving boundaries. Most state-of-the-art methods for such problems, with the notable exception of the immersed boundary method, require unstructured meshes for local discretizations. For irregular domains, mesh gen- eration is still a bottleneckoespecially for three-dimensional problems, problems with moving boundaries and when parallel implementations are used (2). This makes methods based on x ed Cartesian grids attractive for such problems. For certain types of PDEs there is an alternative formulation which is based on inte- gral equations. For example, a constant coefcient elliptic problem can be solved using 1 This work is supported by the National Science Foundation's Knowledge and Distributed Intelligence (KDI) program through grant DMS-9980069.

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