A fast solver for the Stokes equations with distributed forces in complex geometries

We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a black-box fashion; (2) it is second-order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the embedded boundary integral method (EBI), is based on Anita Mayo's work for the Poisson's equation: "The Fast Solution of Poisson's and the Biharmonic Equations on Irregular Regions", SIAM Journal on Numerical Analysis, 21 (1984) 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 equations are discretized by Nyström'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 an N log N algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsity low-rank blocks. 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.