Discretization Principles for Linear Two-Point Boundary Value Problems, II
Tetsuro Yamamoto, Shin’ichi Oishi, Qing Fang · Numerical Functional Analysis and Optimization · 2008
Consider the boundary value problem ℒu ≡ −(pu′)′ + qu′ + ru = f, a ≤ x ≤ b, u(a) = u(b) = 0. Let H ν A ν U = f and be its finite difference equations and piecewise linear finite element equations on partitions , ν = 1, 2,… with , as ν → ∞, where H ν are n ν × n ν diagonal matrices and A ν as well as are n ν × n ν tridiagonal. It is shown that the following three conditions are equivalent: (i) The boundary value problem has a unique solution u ∊ C 2[a, b]. (ii) For sufficiently large ν ≥ ν0, the inverse exists and , ∀ i, j with a constant M > 0 independent of h ν. (iii) For sufficiently large ν ≥ , exists and , ∀ i, j with a constant independent of h ν. It is also shown by a numerical example that the finite difference method with uniform nodes x i+1 = x i + 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.