Approximating surface areas by interpolations on triangulations
For researchers in numerical analysis and geometric modeling, this work removes geometric restrictions for a specific interpolation method, but the result is incremental as it builds on known techniques.
The paper proves convergence of surface area approximations using Crouzeix-Raviart interpolation on triangulations without geometric conditions, whereas Lagrange interpolation requires the maximum angle condition. It provides an alternative proof for Young's classical result on Lagrange interpolation.
We consider surface area approximations by Lagrange and Crouzeix--Raviart interpolations on triangulations. For Lagrange interpolation, we give an alternative proof for Young's classical result that claims the areas of inscribed polygonal surfaces converge to the area of the original surface under the maximum angle condition on the triangulation. For Crouzeix--Raviart interpolation we show that the approximated surface areas converge to the area of the original surface without any geometric conditions on the triangulation.