Operator-Splitting/Finite Element Methods for the Minkowski Problem

The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the Minkowski problem; however, the numerical solution of this problem has been largely left behind, with only a few methods available to achieve that goal. In this article, focusing on the two-dimensional Minkowski problem with Dirichlet boundary conditions, we introduce two solution methods, both based on operator-splitting. One of these two methods deals directly with the Dirichlet condition, while the other one uses an approximation à la Robin of this Dirichlet condition. The relaxation of the Dirichlet condition makes the second method better suited than the first one to treat those situations where the Minkowski equation (of Monge-Ampère type) and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge-Ampère equation discussed by Glowinski et al. [J. Sci. Comput., 81 (2019), pp. 2271-2302]; as such they take advantage of a divergence formulation of the Minkowski problem, which makes it well suited to both a mixed finite-element approximation and the time-discretization via an operator-splitting scheme of an associated initial value problem. Our methodology can be easily implemented on convex domains of rather general shape (with curved boundaries, possibly). The numerical experiments validate both methods, showing that if one uses continuous piecewise affine finite-element approximations of the solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second-order accuracy for the L² and L^\infty norms of the approximation error, where the Minkowski-Dirichlet problem is assumed to have a smooth solution. One can easily extend the methods discussed in this article to address the solution of three-dimensional Minkowski problems.