An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge–Kantorovich optimization

A new cell-area equidistribution method for two-dimensional grid adaptation, based on Monge–Kantorovich optimization (or Monge–Kantorovich optimal transport), is presented. The method is based on a rigorous variational principle, in which the L2L2 norm of the grid displacement is minimized, constrained locally to produce a prescribed positive-definite cell volume distribution. The procedure involves solving the Monge–Ampère equation: A single, nonlinear, elliptic scalar equation with no free parameters, and with proved existence and uniqueness theorems. We show that, for sufficiently small grid displacement, this method also minimizes the mean grid-cell distortion, measured by the trace of the metric tensor. We solve the Monge–Ampère equation numerically with a Jacobian-Free Newton–Krylov method. The ellipticity property of the Monge–Ampère equation allows multigrid preconditioning techniques to be used effectively, delivering a scalable algorithm under grid refinement. Several challenging test cases demonstrate that this method produces optimal grids in which the constraint is satisfied numerically to truncation error. We also compare this method to the well known deformation method [G. Liao, D. Anderson, Appl. Anal. 44 (1992) 285]. We show that the new method achieves the desired equidistributed grid using comparable computational time, but with considerably better grid quality than the deformation method.