Does FOAM converge to exact solution in Laplace equation?
Recently Nisi and me have been working in understanding the nonOrthogonal correction method implemented in FOAM. As have been explained in another threads Over-Relaxed approach is used to correct the solution iteratively for non-orthogonal meshes. We carried out several runnings with the same problem but incresing non-orthogonality. The mesh was simple, a bar with only two triangular (wedge) elements, and different large-height ratio. Since the equation was the Laplace one and temperatures fixed at both extremes, analytical solution is a line and we could compare it with the numerical data. We've found that numerical solution converges but not to the analytical solution, and this phenomenon is so big as the non-orthogonality is. Nisi have found the following errors for different mesh ratios (for 9 nonOrthogonalCorrections):
Mesh ratio %e
2:1 0.1135 %
5:1 0.53375 %
20:1 1.2209 %
Is that interpretation correct? If it is correct is the approach based in any kind of compromise solution or what is the idea behind it (remember that even solution is fairly good, this approach would violate one the cornerstones of numerical analysis --> convergence to analytic solution) ?
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