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

Abstract

A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.

Fourth-order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems

“…These results are only for linear elliptic differential equations with constant coefficients. The similar results were given in [22] for a coupled system of two special semilinear elliptic differential equations with constant coefficients. There is relatively little discussion on the theoretical analysis and the actual implementation of the fourth-order compact finite difference methods applied to nonlinear elliptic differential equations, especially for the case of variable coefficients.…”

“…The main idea of these methods is to increase the accuracy of the standard central finite difference approximation from the second-order to the fourth-order by approximating compactly the leading truncation error terms. In a similar manner, a class of fourth-order compact finite difference methods were proposed in [20][21][22] for some second-order and fourth-order semilinear elliptic differential equations. In all these works (except [22]), the main concern is the construction of the method.…”

“…In a similar manner, a class of fourth-order compact finite difference methods were proposed in [20][21][22] for some second-order and fourth-order semilinear elliptic differential equations. In all these works (except [22]), the main concern is the construction of the method. The corresponding theoretical analysis, such as the existence-uniqueness problem and the error estimate of the numerical solution, was not given.…”

Prediction of Protein Structure Using Surface Accessibility Data

“…Recently, an approach has been presented for the molecular dynamics software XPLOR-NIH using a structure-based metric including the neighboring heavy atoms. [9] Herein, we use a different approach optimized for high-performance and time-efficiency in which we directly use a model structure and map it onto a bit array (Figure 1 b). This simplifies the required computations to simple gridbased operations that are further accelerated by lookup tables.…”

New High Accuracy Non-polynomial Spline Group Explicit Iterative Method for Two Dimensional Elliptic Boundary Value Problems

“…These equations arise very frequently in describing velocity potentials, stationary distribution of temperatures, potential ows, and structural mechanics. Thus, solving this type of equation has been of interest to many researchers [1][2][3][4][5]. We consider the two-dimensional elliptic partial di erential equation of the form: (2) Assume that the boundary conditions are given with su cient smoothness to maintain the order of accuracy in the numerical method under consideration.…”