A Geometrical-mechanical Interpretation of Gradient-weighted Moving Finite Elements

The usual explanation of the gradient-weighted moving finite element (GWMFE) method has been in terms of its variational interpretation. This paper presents a more intuitive geometrical-mechanical interpretation of GWMFE as a balance of forces on the nodes, forces concentrated onto the nodes by the laws of leverage. It also presents significant simplifications in the “internodal viscosity” terms for regularization of the nodal movements, plus some simple “linear internodal tensions” for regularization of the long-term nodal positioning. These simplifications of the regularizations are especially important in two and three space dimensions. One of the generalizations which follows from the geometrical-mechanical interpretation is a promising but still untested second GWMFE formulation for systems of PDEs. The original MFE method is seen to be the small-slope limit of GWMFE under “vertical rescaling.” Reporting on the design and extensive numerical trials of robust and versatile GWMFE systems codes in one and two dimensions is deferred to two forthcoming papers by Carlson and the author [Design and application of a gradient-weighted moving finite element code, Part I, in 1-D, SIAM J. Sci. Comput., to appear] and [Design and application of a gradient-weighted moving finite element code, Part II, in 2-D, SIAM J. Sci. Comput., to appear]. Here only a few illustrative examples are presented involving motion of surfaces by mean curvature, i.e., by surface tension.