Kenta Kobayashi

Kenta Kobayashi, Takuya Tsuchiya

We consider the error analysis of Lagrange interpolation on triangles and tetrahedrons. For Lagrange interpolation of order one, Babuška and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed triangles and tetrahedrons are proved by a method that is a straightforward extension of the original given by Babuška-Aziz.

Kenta Kobayashi, Takuya Tsuchiya

We discuss the error analysis of the lowest degree Crouzeix-Raviart and Raviart-Thomas finite element methods applied to a two-dimensional Poisson equation. To obtain error estimations, we use the techniques developed by Babuška-Aziz and the authors. We present error estimates in terms of the circumradius and the diameter of triangles in which the constants are independent of the geometric properties of the triangulations. Numerical experiments confirm the results obtained.

Kenta Kobayashi, Takuya Tsuchiya

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.

Kenta Kobayashi, Takuya Tsuchiya

We present the error analysis of Lagrange interpolation on triangles. A new \textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates.

Kenta Kobayashi, Takuya Tsuchiya

We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential then the well-known maximum angle condition for convergence of the finite element method. Numerical experiments show that this condition is the best possible. We also point out that the circumradius condition is closely related to the definition of surface area.