Solution of the biharmonic equation on regions with corners

Solution of the biharmonic equation on regions with corners

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Manas Rachh, Yale University
Fine Hall 224

The design of microfluidic devices involves the study of fluid dynamics at small length scales. The behavior of such systems is governed by the Stokes equations (biharmonic equation with gradient boundary conditions), due to the dominant influence of lubrication effects. Typically, micro-channels in microfluidic devices manufactured using planar lithographic techniques have nearly rectangular cross sections and thus tend to have many sharp corners. In such cases, reformulating the governing equation as a boundary integral equation is a natural approach, since this reduces the dimensionality of the problem (discretizing the boundaries alone) and permits high order accuracy in complicated geometries. However, whenever the computational domain has corners, the use of integral equation methods requires particular care as the resulting solutions develop singularities. It is conjectured by Osher (and proven in certain special cases) that the Green’s function for the biharmonic equation on regions with corners has infinitely many oscillations in the vicinity of each corner. In this talk, we show that the solutions of one of the integral equations equivalent to the biharmonic equation can be represented by a series of elementary functions which oscillate with a frequency proportional to the logarithm of the distance from the corner. This representation is used to construct accurate and efficient Nyström discretizations for solving the resulting integral equation with a moderate number of unknowns. We illustrate the performance of our method with several numerical examples.