A fully implicit, nonlinear adaptive grid strategy

We propose an efficient, fully implicit, nonlinear solver for the Beltrami grid generation equation. The Beltrami equation is obtained using Harmonic map theory, and therefore the existence and uniqueness of a solution is guaranteed. The nonlinear solver strategy is based on Newton–Krylov methods, preconditioned here with a multigrid-based method for scalability. Numerical experiments performed for both grid adaptation and grid alignment are presented and demonstrate optimal scaling under grid refinement. We therefore conclude that such a fully nonlinear approach is indeed feasible and efficient.