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

Abstract

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.

A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

“…Despite knowing analytic expressions for these integrals, the expressions for the integrals can be subject to severe roundoff errors and great care has to be taken in their evaluation, as in [8,9] where they develop formulas for these integrals to avoid roundoff error.…”

“…The SGWMFE formulation for systems of partial differential equations was originally proposed by [22] as an alternative formulation of the Gradient Weighted Moving Finite Element (GWMFE) method, which was developed in detail in [8,9] by Carlson and Miller for one and two-dimensional problems. The paper [8] reports on the design and implementation of a robust and versatile GWMFE code in one dimension applied to various PDEs and PDE systems.…”

“…There they found that GWMFE efficiently produces accurate results for problems which form steep moving fronts. The corresponding two-dimensional paper [9] does the same as the one-dimensional paper including the additional application problems: 1) non-linear arsenic diffusion, 2) the Buckley-Leverett "black oil" equations, and 3) motion by mean curvature which was implemented in the one-dimensional case in the paper [22] by K. Miller. The results therein show that the method is intended for "problems with sharp moving fronts where one needs to resolve the fine-scale structure of the front to compute the correct answer" greatly improving on the first Moving Finite Element (MFE) method developed originally by K. Miller and R.N.…”

Lagrangian and moving mesh methods for the convection diffusion equation

“…One method of adapting the mesh to the solution of an evolution equation is to let the mesh evolve with the solution [5,19,20]. For convection dominated problems it is natural to let the mesh points flow along the streamlines (characteristics) of (an approximation of) the velocity.…”

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

“…An important alternative, relating to arclength, is to take w = 1/(1+ |∇u| 2 ) and carry out the integration over arclength, which is known as the gradientweighted form of MFE (GWMFE) and is described in [37,38]. The further generalization of this approach to systems of PDEs is considered in [101] and includes a so called "string gradient-weighted" formulation (SGWMFE) which is discussed in more detail in [143].…”