the author and R. Miller introduced powerful new numerical methods, for dealing with problems whose solutions develop sharp transition layers or "near-shocks". These methods allow many nodes automatically to concentrate in the critical regions and move with them. Part presented the theoretical and computational details of the scheme, along with some trial runs for Burgers' equation which showed that the nodes do concentrate and move as desired; but the conclusiveness of the results was marred by failure to debug a working Newton's method for our stiff ODE solver, with a consequent restriction to very tiny At's. The present paper, Part II, presents the results of later computations and developments which show quite conclusively the power and efficiency of these new me/hods. Improvements include a working implicit stiff equations solver, addition of slight "internodal spring forces" to keep the nodes at least slightly separated, and trial of the method on a wide variety of hyperbolic and parabolic PDE's.
In part I the authors reported on the design of a robust and versatile gradientweighted moving finite element (GWMFE) code in one dimension and on its application to a variety of PDEs and PDE systems. This companion paper does 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. The many potential pitfalls in the design of GWMFE codes and the special features of the implicit onedimensional (1D) and 2D codes which contribute to their robustness and efficiency are discussed at length in part I; this paper concentrates on issues unique to the 2D case. Brief explanations are given of the variational interpretation of GWMFE, the geometrical-mechanical interpretation, simplified regularization terms, and the treatment of systems. A catalog of inner products which occur in GWMFE is given, with particular attention paid to those involving second-order operators. After presenting an example of the 2D phenomenon of grid collapse and discussing the need for long-time regularization, the paper reports on the application of the 2D code to several nontrivial problemsnonlinear arsenic diffusion in the manufacture of semiconductors, the drift-diffusion equations for semiconductor device simulation, the Buckley-Leverett black oil equations for reservoir simulation, and the motion of surfaces by mean curvature.
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