Numerical differentiation of approximated functions with limited order-of-accuracy deterioration (Q2903051)

In many numerical applications such as the numerical solution of (partial) differential equations, the question of approximating and differentiating the underlying functions numerically arises. Since certain orders of the solution methods (e.g., of the partial differential equation) are desired, there should be no or minimal loss of accuracy during the process of the numerical differentiation. This is the theme of the current paper where new methods are proposed for exactly this purpose; the authors use Chebyshev approximations (Chebyshev interpolants) to minimise the loss of accuracy during the differentiation process. The errors that can arise in these new methods are analysed and upper bounds are provided. To show the usefulness of the ideas, numerical examples, domain decomposition approaches and comparisons with radial basis function methods are given.