In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in $\mathbb{R}^3$. Previous research used a restriction to rotationally symmetric domains to reduce shape optimization problems to a two-dimensional setting. In the current research, two approaches for the approximation in $\mathbb{R}^3$ are considered. First, a notion of discrete convexity allows for a nearly convex approximation with polyhedral domains. An alternative approach is based on the recent observation that higher order finite elements can approximate convex functions conformally. As a second approach these results are used to approximate optimal convex domains with isoparametric convex domains. The proposed algorithms were tested on shape optimization problems constrained by a Poisson equation and both algorithms achieved similar results.

翻译：在带偏微分方程约束的凸域优化问题中，$\mathbb{R}^3$内凸域的逼近存在数值困难。先前的研究通过限制旋转对称域将形状优化问题简化为二维情形。当前研究考虑了两种$\mathbb{R}^3$中的逼近方法：首先，基于离散凸性概念，利用多面体域实现近似凸逼近；其次，基于高阶有限元可共形逼近凸函数的最新发现，采用等参凸域逼近最优凸域。将所提算法应用于泊松方程约束的形状优化问题测试，两种算法取得了相似结果。