Discretization Principles for Linear Two-Point Boundary Value Problems

Abstract

Consider the boundary value problem ℒu ≡ −(pu′)′ + qu′ + ru = f, a ≤ x ≤ b, u(a) = u(b) = 0. Let HAU = f be its finite difference equations on ▵: a = x₀ < x₁ <···< x_n+1 = b, h_i = x_i − x_i−1, h = max h_i, where H = diag(ω₁,…, ω_n), ω_i = (h_i + h_i+1)/2, A is a tridiagonal n × n matrix. It is then shown that if the problem has a unique solution u, then A is nonsingular for sufficiently small h. Let and G(x, ξ) be the Green function for ℒ. Then as h → 0 and there exists a constant M > 0 independent of h such that ∀i, j, which generalize Tikhonov–Samarskii's results. Similar results hold for the piecewise linear finite element equations.

Keywords:

• Discretization principles
• Finite difference methods
• Finite element methods
• Two-point boundary value problems

AMS Subject Classification:

• 65L10
• 65L12
• 65L60

ACKNOWLEDGMENT

The author thanks Professor Tsuchiya of Ehime University for bringing the existence of reference [5] to the author's attention.