Abstract
Two problems that include corner singularities are considered. The first concerns the flow of a viscous fluid in a channel driven by a constant pressure gradient, when the velocity satisfies a two-dimensional Poisson equation. The second is Stokes flow in a two-dimensional region when the stream-function satisfies the biharmonic equation. For both problems the boundaries of the domains contain corners. For corner angles greater than some critical value, the stress or the vorticity is singular. Using both a formal analysis and numerical results, we show that numerical approximations for the stream-function and velocity, obtained by using standard second-order finite difference methods, still converge to the exact solutions despite the corner singularities. However, the convergence rate is lower than second-order and the deterioration in the accuracy is not local, i.e., not confined to the corner. On the other hand, even though the vorticity solution of the Stokes problem does not converge, it diverges only locally. At a finite distance from the corner, the vorticity converges with the same rate as the stream-function. Adaptive methods for improving the accuracy are also discussed.
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Huang, H., Seymour, B.R. Finite Difference Solutions of Incompressible Flow Problems with Corner Singularities. Journal of Scientific Computing 15, 265–292 (2000). https://doi.org/10.1023/A:1011138516712
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DOI: https://doi.org/10.1023/A:1011138516712