Error Analysis in Bivariate Spline Interpolation. This was done for differentiable splines of degree at least four defined on domains divided into subrectangles with one diagonal. Romberg Type Cubature over Arbitrary Triangles. The use of barycentric coordinates and the interpretation of the trapezoidal rule as the integral over an interpolating linear spline function. Moreover, also numerical integration over convex polyhedra in higher dimensional spaces can be accomplished without difficulties.
Numerical examples show the stability and efficiency of the algorithm.
Numerical Solution of Fractional Order Differential Equations by Extrapolation. We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm, possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples.