The Scalar Homotopy Method for Solving Non-Linear Obstacle Problem

Abstract

In this study, the nonlinear obstacle problems, which are also known as the nonlinear free boundary problems, are analyzed by the scalar homotopy method (SHM) and the finite difference method. The one- and two-dimensional nonlinear obstacle problems, formulated as the nonlinear complementarity problems (NCPs), are discretized by the finite difference method and form a system of nonlinear algebraic equations (NAEs) with the aid of Fischer-Burmeister NCP-function. Additionally, the system of NAEs is solved by the SHM, which is globally convergent and can get rid of calculating the inverse of Jacobian matrix. In SHM, by introducing a scalar homotopy function and a fictitious time, the NAEs are transformed to the ordinary differential equations (ODEs), which can be integrated numerically to obtain the solutions of NAEs. Owing to the characteristic of global convergence in SHM, the restart algorithm is adopted to fasten the convergence of numerical integration for ODEs. Several numerical examples are provided to validate the efficiency and consistency of the proposed scheme. Besides, some factors, which might influence on the accuracy of the numerical results, are examined by a series of numerical experiments.