Diagonal Dominance and Positive

We dedicate this paper to Professor Ron Mitchell who has been an inspiration to many generations of numerical analysts. In addition his generous spirit has infused our discipline with a sense of positive endeavour. We examine whether three diierent upwind schemes for the positive definite advection diiusion problem can yield indeenite coeecient matrices. In particular, in line with the increasing use of adaptive meshes, we are concerned with discretisations on arbitrary grids but only in one dimension. We show that indeenite coeecient matrices can arise from certain approaches: this can present diiculties with eecient iterative solution techniques which might be required for corresponding approximations in higher dimensions.