Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
A survey on various computational techniques for nonlinear elliptic boundary value problems
Advances in Engineering Software
A fourth-order compact finite difference method for higher-order Lidstone boundary value problems
Computers & Mathematics with Applications
Journal of Computational and Applied Mathematics
SIAM Journal on Numerical Analysis
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This paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth-order elliptic boundary value problems in a two-dimensional domain. The boundary value problem is discretized by the finite difference method, and two iterative processes, called block Jacobi and block Gauss-Seidel monotone iterations, are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration. It is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi-monotone nondecreasing. A sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system. Similar results are obtained for quasi-monotone nonincreasing functions. An analytical comparison relation between the block Jacobi and block Gauss--Seidel monotone iterations is obtained. It is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero. Numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy. Also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations.