1) I implemented the Middlecoff-Thomas procedure for a 2D elliptic grid generator; the grid that outcomes loses the orthogonality at the boundaries! Is it ordinary???

2) Which kind of linear solver do you use to invert the matrix of an elliptic grid generator?

If the grid point number is huge, Iterative methods are better than direct ones. I use point Gauss-Seidal, or line Gauss-Seidel, both with a multigrid acceleration. It seems work well for elliptic grid generator.

> 1) I implemented the Middlecoff-Thomas procedure
: for a 2D elliptic grid generator; the grid that
: outcomes loses the orthogonality at the boundaries!
: Is it ordinary???

Yes, this is the expected behavior. Thomas-Middlecoff primarily impacts clustering on the grid's interior so that it mimics boundary clustering. A seconary effect is smoothness. There is little in T-M to control orthogonality at the boundaries. For that you need a technique like Steger-Sorenson or von Lavante - Hilgenstock - White.