Design and Application of a Gradient-Weighted Moving Finite Element Code II: in Two Dimensions

  • Authors:

  • Neil N. Carlson;Keith Miller

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Scientific Computing
  • Year:
  • 1998

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Abstract

In part I the authors reported on the design of a robust and versatile gradient-weighted 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 one-dimensional (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 problems---nonlinear 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.