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 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 method uses an approximation of this Dirichlet condition. This relaxation of the Dirichlet condition makes this second method better suited than the first one to treat those situations where the Minkowski and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge-Amp\`{e}re equation discussed by Glowinski et al. (Journal of Scientific Computing, 79(1), 1-47, 2019); as such they take advantage of a divergence formulation of the Minkowski problem, well-suited to a mixed finite element approximation, and to the 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 we performed validate both methods and show that if one uses continuous piecewise affine finite element approximations of the smooth solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second order accuracy for the $L^2$ and $L^{\infty}$ error. One can extend easily the methods discussed in this article, to address the solution of three-dimensional Minkowski problem.

翻译：凸体的经典明科夫斯基问题深刻影响了微分几何的发展。过去几十年间，针对明科夫斯基问题的解已发展出丰富的数学理论，然而该问题的数值求解仍相对滞后，现有方法寥寥无几。本文聚焦带狄利克雷边界条件的二维明科夫斯基问题，介绍了两种基于算子分裂的求解方法。其中一种方法直接处理狄利克雷条件，另一种则采用该条件的近似形式。这种对狄利克雷条件的松弛处理使得第二种方法在明科夫斯基条件与狄利克雷条件不兼容时比第一种方法更具优势。两种方法均为Glowinski等学者（Journal of Scientific Computing, 79(1), 1-47, 2019）提出的经典蒙日-安培方程求解方法的推广，因此利用了明科夫斯基问题的散度公式——该公式适用于混合有限元逼近，并通过算子分裂格式实现相关初值问题的时间离散化。我们的方法可轻松应用于形状较为一般（可能具有弯曲边界）的凸域。数值实验验证了两种方法的有效性，结果表明：若采用连续分段仿射有限元逼近明科夫斯基问题的光滑解及其三阶二阶导数，这两种方法在$L^2$和$L^{\infty}$误差上均能达到近二阶精度。本文讨论的方法可轻松扩展至三维明科夫斯基问题的求解。