Optimal mass transport for higher dimensional adaptive grid generation

In this work, we describe an approach for higher dimensional adaptive grid generation based on solving the L2 Monge–Kantorovich problem (MKP) which is a special case of the classical optimal mass transportation problem. Two methods are developed for computing the coordinate transformation used to define the grid adaptation. For the first method, the transformation is determined by solving a parabolic Monge–Ampère equation for a steady state solution. For the second method, the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the L2 MKP. Several numerical experiments are presented to demonstrate the performance of the MKP methods and to compare them with some related adaptive grid methods. The experimental results demonstrate that the MKP methods show promise as effective and reliable methods for higher dimensional adaptive grid generation.