Second kind integral equation formulation for the modified biharmonic equation and its applications

A system of Fredholm second kind integral equations (SKIEs) is constructed for the modified biharmonic equation in two dimensions with gradient boundary conditions. Such boundary value problem arises naturally when the incompressible Navier-Stokes equations are solved via a semi-implicit discretization scheme and the resulting boundary value problem at each time step is then solved using the pure stream-function formulation. The advantages of such an approach (Greengard and Kropinski, 1998) [14] are two fold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. Our construction, though similar to that of Farkas (1989) [10] for the biharmonic equation, is modified such that the SKIE formulation has low condition numbers for large values of the parameter. We illustrate the performance of our numerical scheme with several numerical examples. Finally, the scheme can be easily coupled with standard fast algorithms such as FFT, fast multipole methods (Greengard and Rokhlin, 1987) [15], or fast direct solvers (Ho and Greengard, 2012; Martinsson and Rokhlin, 2005) [17], [25] to achieve optimal complexity, bringing accurate large-scale long-time fluid simulations within practical reach.