Discretization principles for linear two-point boundary value problems, II

Consider the boundary value problem Lu-(pu')'+qu'+ru=f, a≤x≤b, u(a)=u(b)=0. Let HνAνU=f and [image omitted] be its finite difference equations and piecewise linear finite element equations on partitions [image omitted], ν=1, 2,... with [image omitted], [image omitted] as ν, where Hν are nνnν diagonal matrices and Aν as well as [image omitted] are nνnν tridiagonal. It is shown that the following three conditions are equivalent: (i) The boundary value problem has a unique solution uC2[a, b]. (ii) For sufficiently large νν0, the inverse [image omitted] exists and [image omitted], i, j with a constant M0 independent of hν. (iii) For sufficiently large ν[image omitted], [image omitted] exists and [image omitted], i, j with a constant [image omitted] independent of hν. It is also shown by a numerical example that the finite difference method with uniform nodes xi+1=xi+h, 0≤i≤n, h=(b-a)/(n+1) applied to the boundary value problem with no solution gives a ghost solution for every n.