We derive a numerical approximation of the Laplace-Beltrami operator on compact surfaces embedded in $\mathbb{R}^3$ with an axial symmetry. To do so we use a noncommutative Laplace operator defined on the space of finite dimensional hermitian matrices. This operator is derived from a foliation of the surface obtained under an $S^1$-action on the surface. We present numerical results in the case of the sphere and a generic ellipsoid.

翻译：本文推导了嵌入 $\mathbb{R}^3$ 且具有轴对称性的紧致曲面上拉普拉斯-贝尔特拉米算子的数值逼近方法。为此，我们使用定义在有限维厄米矩阵空间上的非交换拉普拉斯算子。该算子源于曲面在 $S^1$ 作用下所获得的叶状结构。我们展示了球面及一般椭球面情形的数值计算结果。