On the norm equivalence of singularly pertubed elliptic difference operators

摘要

Consider the system of linear algebraic equations LK,hU=f which arises from the finite-difference discretization of the singularly pertubed elliptic operator ∷LKu∷= {∂∂xa∂u∂x+∂∂xb∂u∂y+∂∂yb∂u∂x+∂∂yc∂u∂y} + K° {d∂u∂x+ e∂u∂y} + Kpu where 1≪K<∞, 0⩽σ⩽1. In this work we are concerned with preconditioning operators L̃K,h so that p](1)the condition number of (L̃K,h−1LK,h and the condition number of LK,h(L̃K,h)−1 are bounded independently of h, and p](2)there are bounds on the growth of these condition numbers as a function of K.