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This paper develops a nested iteration algorithm to solve time-dependent nonlinear systems of partial differential equations. For each time step, Newton's method is used to form approximate solutions from a sequence of nested spaces, where the resolution of the approximations increases as the algorithm progresses. Nested iteration results in most of the iterations being performed on coarser grids, where minimal work is needed to reduce error to the level of discretization error. The approximate solution on a given coarse grid is interpolated to a refined grid and is used as an initial guess for the problem posed there. The approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem due to the rapid convergence of Newton's method near a solution. The paper develops an algorithm that attempts to optimize accuracy-per-computational-cost on each grid, so that essentially no unnecessary work is done on any grid. The nested iteration algorithm is then applied to a reduced two-dimensional model of the incompressible, resistive magnetohydrodynamic (MHD) equations. Using this algorithm on the MHD equations in the context of a first-order system least squares finite element discretization and algebraic multigrid to solve the linearized systems, instabilities in a model tokamak fusion reactor are simulated. Numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30-80 fine-grid relaxation sweeps per time step.

Nested Iteration and First-Order System Least Squares for Incompressible, Resistive Magnetohydrodynamics