From Stepwise Integrations and Low-rank Updates to a Pseudospectral Solution Operator Matrix for the Helmholtz Operator $\frac{d}{dx} a(x) \frac{d}{dx} + c(x)$

Abstract

In this study we propose a construction framework utilizing stepwise integrations and the Sherman-Morrison-Woodbury formula to seek pseudospectral integration preconditioning matrices for differential operators. We illustrate this framework through formulating an inverse matrix for the Helmholtz differential operator of the form $\frac{d}{dx} a(x) \frac{d}{dx} + c(x)$. Numerical experiments were conducted to examine the performance of the derived operator. The results show that the inverse matrix is an effective solution operator to numerically solve general second order differential equations.