A two-grid method for the $C^0$ interior penalty discretization of the Monge-Ampère equation
For researchers solving Monge-Ampère equations numerically, this provides a more efficient solver, though it is an incremental application of existing two-grid techniques.
The paper proposes a two-grid method for solving the nonlinear system from a C0 interior penalty discretization of the Monge-Ampère equation, achieving computational efficiency by solving on a coarse mesh and using one Newton iteration on a fine mesh. Numerical experiments confirm efficiency gains over standard Newton's method.
The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.