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The Monge-Kantorovich grid generation and adaptation scheme of [Delzanno et al., J. Comput. Phys., 227 (2008), pp. 9841-9864] is generalized from a variational principle based on $L_{2}$ to a variational principle based on $L_{p}$. A generalized Monge-Ampère (MA) equation is derived and its properties are discussed. Results for $p1$ are obtained and compared in terms of the quality of the resulting grid and a measure of computational performance. We conclude that for the grid generation application, the formulation based on $L_{p}$ for $p$ close to unity can lead to serious problems associated with the boundary. On the other hand, $p\gg2$ also leads to worse quality grids and performance. Thus, it is concluded that $p=2$ produces the best quality grids, particularly in terms of mean grid cell distortion. Furthermore, the Newton-Krylov methods used to solve the generalized MA equation perform best for $p=2$.

Generalized Monge–Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$