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Numerical Solutions on Tangled Mesh:
 The Cauchy-Riemann equations for the velocity potential $\phi$ and the stream function $\psi$ are solved by the discrete least-squares residual-minimization scheme: the solutions and the nodal positions are computed by minimizing the residuals. The exact solutions are available as shown in Figures 3 and 4 below. The initial regular triangular mesh (Figure 14) was adapted by the residual-minimization and turned into as shown in Figure 15. It is tangled due to aggressive mesh movement, with negative volumes created. But the method converges simultaneously computing the solution on that grid with no problems. The results are shown in Figures 8 and 9 below. The solution contours look rather reasonable on such a bad mesh, don't they? If you don't like the resulting grid, locally re-triangulate it (remove any negative volume), and you'll have a reasonable solution on a valid grid. It is nice that the computation never fails with negative volumes. [Note: the solution and the grid points were solved simultaneously; the grid converged first and then the solution converged later on that grid.] See [ PhD Thesis ], Section 5.1, page 73, for the residual-minimization scheme.
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