This paper is concerned with the numerical solution of singularly perturbed two point boundary value problems
\[ \varepsilon y''(x)+ p(x) y'(x)= f(x),\quad x \in (0,1), \;y(0)=y(1)=0,\tag{1} \] where the functions \(p\) and \(f\) are sufficiently smooth and \( p(x) \geq p_0 >0, x \in I= [0,1]\). Since the problem has a unique solution with a boundary layer at \(x=0\), the authors consider a piecewise uniform Shishkin mesh in which the interval \([0,1]\) is divided into two subintervals \([0,\tau]\) and \([\tau,1]\) and \(N/2\) grid points are uniformly placed in each subinterval. Then, the solution \(y(x)\) of (1) is approximated by a quadratic spline \( u \in C^1(I)\) and the coefficients of \(u\) at each \([x_i, x_{i+1}]\) are determined by imposing that \(u\) satisfies some comparison problem at \( x_{i+1/2}\) as well as the boundary conditions.
In this context, the authors construct the discrete Green function and, under suitable assumptions, derive uniform bounds for it. Next, after obtaining a bound of the consistency error, they prove an \(\varepsilon\)-uniform convergence bound: \( \max_{i} | y(x_i) -u_i | \leq M N^{-2} (\ln N)^2 \) with a constant \(M\) independent of \(\varepsilon\), for \(N\) sufficiently large and \( y \in C^4(I)\). The paper ends with the numerical results obtained for a test equation that confirm the order of convergence predicted by the theory.