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

Abstract

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.

Keywords:

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

AMS Subject Classification:

• 65L10
• 65L12
• 65L60

ACKNOWLEDGMENTS

The authors are grateful to Professor M.Z. Nashed, University of Central Florida, for his helpful comments during the preparation of this paper. The authors thank also to the referee for his useful comments, by which this paper has been improved.