Inverse matrices for pseudospectral differentiation operators in polar coordinates by stepwise integrations and low-rank updates

Abstract

Spectral/pseudospectral integration preconditioning matrices are useful tools for solving differential equations involving pure differential operators dm/dxm. In this study we construct well-conditioned inverse pseudospectral matrices for the basic differential operator and the mixed differential operator ddra(r)ddr on Gauss-Radau-Legendre points based on stepwise integrations and low-rank updates. The inverse matrices can be used either as a solution operator or an effective preconditioner for variable coefficient differential equations of first and second order in polar coordinates. Numerical experiments were conducted and we observed the performance of the inverse operator as expected.