On an auto-correction phenomenon of the Krylov-Gottlieb-Eckhoff method
Arnak Poghosyan · IMA Journal of Numerical Analysis · 2010
We consider the so-called Krylov–Gottlieb–Eckhoff (KGE) approximation of a function f with a discontinuity at a known point. This approximation is based on certain corrections associated with the jumps in the first q derivatives of f. The approximation of the exact jumps is accomplished by the solution of a system of linear equations. We show that, in the regions where the period-2 extension of the approximated function is smooth, the KGE method with approximate values of the jumps converges faster compared with the case where the exact values are used. We call this accelerated convergence the auto-correction phenomenon, which was discovered in the past by numerical experiments. The paper presents a theoretical explanation of the phenomenon with numerical illustrations.