Two approaches to the difference method of solving Neumann's problem in a rectangular region

Abstract

TWO approaches, based on the idea of regularization, are presented to the construction of a stable approximation of the normal solution of boundary value problems for differential equations on a spectrum, by the example of Neumann's problem for Poisson's equation with a homogeneous boundary condition. In the difference method of solving boundary value problems for differential equations on a spectrum (that is, those for which the corresponding homogeneous problem has a non-zero solution) the difference equation usually obtained is usually either not solvable at all, or is solvable but not uniquely. The solvability condition of the difference problem, consisting of the orthogonality of its right side to some subspace Z 0 connected with the operator of the difference problem, is of an unstable nature, destroyed for small perturbations of the initial data of the problem. Moreover, a change of the data of the difference problem for which the solvability conditions are satisfied, is possible only on the assumption of absolute accuracy of the calculations on the computer. Below two approaches are presented, based on the regularization idea [1], which do not require prior information about the nature of Z 0 and make it possible to construct a stable approximation to the normal solution [2] of a problem of the class indicated, by the example of Neumann's problem for Poisson's equation, assuming its solvability, with a homogeneous boundary condition D = {0 < x v < l v , v = 1, 2} the continuous problem (that is, by the example of the boundary value problem Δ = −f(x), ( δu δn ) ¦ Γ = 0) . We use a known difference operator \\ ̄ gL from [3], which on the solution of the problem has an approximation error ¦h¦ 2 = h 1 2 + h 2 2 , is selfconjugate and positive-definite in the sense of the scalar product [.,.], being the network analog of the quadratic trapezium formula in D (see [4]). A number of papers have been devoted to the difference method of solving the Neumann problem in a rectangular region. In [3], giving the projection of the solution onto the zero proper subspace of the difference operator and constructing the right side in such a way that the difference problem was solvable, the author, by the method of prior estimates, proved convergence to a solution possessing the same projection on the proper subspace of the continuous problem in the net norm W 2 (1), and in [5] the uniform convergence of the difference problem on the introduction of a generalized Green's problem was proved, the rate of convergence in both problems being identical with the order of approximation. In [6] a stable method of finding the normal solution of a difference problem, which approximates the Neumann problem for Poisson's equation with an error O(¦h¦ 2) within and O(¦h¦) on the boundary was proposed, on the assumption that the difference problem is solvable, the convergence to the solution of the continuous problem not being considered. In [7] a scheme for the solution of Laplace's equation was constructed and convergence at the rate O(¦h¦ 2¦ln h¦) was demonstrated.