The circumradius condition and its application

The analysis of the interpolation error is particularly important for the error analysis of the finite element methods. In the previous paper, we proved that the finite element solution converges to an exact solution if the maximum circumradius of the triangular elements converges to zero. We call such situation “circumradius condition” and claimed that the circumradius condition is more essential than the well-known maximum angle condition. It is considered that the better finite element solution can be obtained by using the mesh division consists of “good” triangles. However, the generation of such mesh division is time consuming task within the simulation process of the finite element method. On the other hand, the efficient algorithm is known for computing Delaunay triangulation. However, the mesh division produced by Delaunay triangulation sometimes contains collapsed triangles. In this paper, we will introduce “circumradius condition” and show that the efficient error estimate can be obtained by the circumradius condition with Delaunay triangulation.