Discretization Principles for Linear Two-Point Boundary Value Problems, II

Consider the boundary value problem u ≡ −(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.

Keywords

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

AMS Subject Classification

• 65L10,
• 65L12,
• 65L60