Solving Time-Dependent Problems With Finite Element Method Of Lines

The extended Kantorovich method is a semi-analytical method. A number of applications have shown some remarkable advantages in this method, such as high accuracy, rapid convergence, free initial solution choice, fewer terms, etc. Encouraged by the success of the extended Kantorovich method for space-domain problems, the present study makes a substantial extension of this method to the time-dependent problems. The whole study is composed of four major parts as follows.1. Based on Galerkin method, a new method of solving differential equation which is not self-adjoint using the finite element method of lines is proposed. This method is an extension of conventional Ritz method, and a number of typical numerical examples show that the method is very successful.2. Based on the Galerkin method, jointly with the idea of the extended Kantorovich method, the finite element method of lines is extended to time-dependent problems, and further to nonlinear problems. The basic idea and formulation are described and derived in detail, and the corresponding programs have been developed. Numerical results show that the proposed method produces extremely high accuracy and gains substantial efficiency.3. Based on the Gurtin variational principle, multi-term trial function is proposed for solving two-dimensional problems of linear elastodynamics. A class of two-dimensional problems are solved and a number of typical numerical examples are given to show that the proposed trial function is highly valid and accurate.4. In addition, two additional methods are discussed, and numerical results show that the new methods have many advantages and worth further studying.