Design and application of a gradient-weighted moving finite element code I: In one dimension

This paper reports on the design of a robust and versatile gradient-weighted moving finite element (GWMFE) code in one dimension and its application to a variety of difficult PDEs and PDE systems. A companion paper, part II, will do the same for the two-dimensional (2D) case. These moving node methods are especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts. Brief explanations are given of the variational interpretation of GWMFE, the geometrical-mechanical interpretation, simplified regularization terms, and the treatment of PDE systems. There are many possible pitfalls in the design of GWMFE codes; section 5 discusses special features of the implicit one-dimensional (1D) and 2D codes which contribute greatly to their robustness and efficiency. Section 6 uses a few simple examples to illustrate the workings of the method, some difficulties, and reasons for the standard choices of the internodal viscosity regularization coefficient. Section 7 reports numerical trials on several more difficult PDE systems. Section 8 discusses the failure of the method on certain steady-state convection problems. Section 9 describes a simple nonlinear "Krylov subspace" accelerator for Newton's method, a routine which greatly decreases the number of Jacobian evaluations required for our stiff ODE solver.