Numerical methods for fourth‐order nonlinear elliptic boundary value problems

Abstract

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth-order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence-comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two-point boundaryvalue problem with known analytical solution are given.

Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems

“…For comparison, we also solve (6.1) by the standard finite difference method (SFD) as in [16,17]. This method leads to a system of nonlinear algebraic equations of the form (6.4) with the matrices A = tridiag(−I, A 1 , −I ),…”

“…Recently, much attention has been also paid to certain fourth-order elliptic boundary value problems in multiple dimensions (cf. [7,[14][15][16][17]21]). In this paper, we focus on the following two-dimensional fourth-order nonlinear elliptic boundary value problem: ∆(k(x, y)∆u) = f (x, y, u, ∆u), (x, y) ∈ Ω , u(x, y) = g(x, y), ∆u(x, y) = g * (x, y), (x, y) ∈ ∂Ω , (1.2) where Ω is a rectangular domain or a union of rectangular domains, and ∆ is the Laplacian operator.…”

“…[4,6,12]). Recently, a finite difference method was proposed in [16,17], for a class of nonlinear fourth-order elliptic boundary value problems including (1.2), by using the standard second-order finite difference approximation and three monotone iterations for resolving the resulting discrete systems with the quasimonotone function f (·, ·, u, v).…”

Upper and Lower Solution Methods for Fully Nonlinear Boundary Value Problems

“…There have been numerous applications of the monotone method to a problem similar to the fourth order problem we consider; the recent work by Ma et al [13] provides an excellent account of the method with a good set of references; in a closely related work, Bai [5] first applies a new form of a maximum principle and then relaxes some monotonicity hypotheses. Eloe and Islam [12] have applied the monotone method to a very closely related impulse problem and Pao [26] has recently applied the monotone method to a related BVP for a fourth order elliptic partial differential equation. In each of these works [5,12,13,26], there is no dependence on odd order derivatives.…”

“…Eloe and Islam [12] have applied the monotone method to a very closely related impulse problem and Pao [26] has recently applied the monotone method to a related BVP for a fourth order elliptic partial differential equation. In each of these works [5,12,13,26], there is no dependence on odd order derivatives.…”