Study on finite element exterior calculus

给出了三角元线性插值的“圆半径条件”。我们没有使用经过验证的数值计算来证明它。然后我们发现圆周半径条件与表面积的定义密切相关。也就是说，我们证明了如果一个曲面具有一定的规则性，且切曲面满足圆周条件，则切曲面的面积收敛于该曲面的面积。由于该定理可以应用于“施瓦茨灯笼”，因此该定理是关于三角几何的“最佳可能”。

We have found “the circumradius condition” for linear interpolation on triangular elements. We proved it without using validated numerical computation. We then found that the circumradius condition is closely related to the definition of surface area. That is, we proved that if a surface has a certain regularity and inscribed surfaces satisfy the circumradius condition, the area of inscribed surfaces converges to that of the surface. Since the theorem can be applied to “Schwarz’s lantern”, the theorem is “best-possible” with respect to geometry of triangulation.