DOC: iterative linear solution for the Newton decrement of steady Navier-Stokes

[gdmcbain]

In reviewing the example of forced convection #162, some suggestions were made to reduce the five-minute run-time. These concern the Navier–Stokes solver rather than forced convection per se, so new issues are being launched to take them up. The first suggestion was reverse Cuthill–McKee reordering #168; the second was:

Another idea is to save LU-decomposition of self.S and use GMRES with the initial guess from solve(L, solve(U, rhs)) - I assume rhs changes each time. That would require incomplete LU preconditioning based on self.S + block_diag(...).

The Navier–Stokes solver in #145 (reused in #162) has two levels of iteration. Outside is the natural parameter continuation in the Reynolds number; for that, the initial guess can be obtained from the solution of the Stokes problem. Inside is the Newton iteration for the nonlinearity; for that, the initial guess is naturally supplied by the solution at the previous Reynolds number, or by extrapolation, ‘first-order continuation’, &c.

The suggestion involves another level of iteration, for the decrement at each Newton step. The equation is [S + Re N' (u)] du = -f (u) = -{S u + Re N (u)}, currently solved directly for each new u with skfem.utils.solve; i.e., skfem.utils.solver_direct_scipy and scipy.sparse.linalg.solve.

[gdmcbain]

As in

In this chapter, we bring together our ideas underlying fast solvers for Stokes and convection–diffusion problems to develop effective strategies for solving the discrete systems that arise in the solution of the Navier–Stokes equations.

• Elman, H. C., Silvester, D. J., & Wathen, A. J. (2014) ‘Solution of discrete Navier–Stokes problems’, ch. 9 of Finite Elements and Fast Iterative Solvers : with Applications in Incompressible Fluid Dynamics. Oxford University Press

cited in gdmcbain/scikit-fem#1, the preconditioners and iterative schemes for the Newton steady Navier–Stokes equation should follow those for simpler problems.

In turn (Elman et al, ch. 4), schemes for the Stokes problem are generally developed on top of those for the Poisson problem. (With the big addition that Stokes means mixed finite elements and a saddle-point problem.)

I'll leave this open, but defer working on it till after those simpler problems have been treated.

[gdmcbain]

The Stokes problem is treated in PR #342.