Study on finite element exterior calculus

我们找到了三角形单元上线性插补的“外接半径条件”。我们在没有使用经过验证的数值计算的情况下证明了它。然后我们发现，外接半径条件与表面积的定义密切相关。也就是说，如果一个曲面具有一定的正则性，且内接曲面满足外接半径条件，则内接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”，因此该定理对于三角剖分的几何来说是“最佳可能的”。

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.