We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.
翻译:我们提出了一个用于估算两维和三维平滑函数的内推误的一般理论。 在我们的理论中,内推误以简单x和几何参数的直径为界。 在二维的情况下,我们的几何参数相当于三角形的圆形。在三维的情况下,我们的几何参数也代表四面线的平面。通过引入几何参数,新获得的误差估计可适用于违反最大矩形条件的情况。