Moving Finite Elements. II

Abstract

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.

A boundary integral approach to unstable solidification

“…Their numerical results show these estimates to be sharp. Alexander et al [2] have applied Miller's moving finite elements [78] to the enthalpy formulation. Review articles on the subject are [33] , [25] and [76]; these articles also describe many methods which we have omitted.…”

A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines

“…Unfortunately, very few, if any, moving-grid software packages, generally applicable up to nearly the same level of efficiency, robustness, and reliability as conventional packages, are available yet, even for the relatively simple l D case. Admittedly, for an interesting variety of difficult example problems, various adaptive techniques have been shown to be potentially very efficient, a prominent example being the moving-finite-element method invented by Miller and his co-workers [6,10,16,17,18,19]. However, most, of the techniques, including the moving-finite-element method, require some form of tuning to ensure that the automatic choice of the changing space nodes is safely governed.…”

An adaptive moving grid method for one-dimensional systems of partial differential equations

“…We emphasize that this front tracking is achieved automatically by the algorithm and not via a user-supplied coordinate transformation. This capability is shared by other moving grid methods (e.g., the moving finite-element method [17,18] and Petzold's finite-difference method [20]). …”

“…Concerning the grid determination, our algorithm can be classified as belonging to the class of methods which are "intermediate" between the static regridding methods, where nodes remain fixed for intervals of time [ 14,[22][23][24], and continuously moving grid methods, where the node movement and the PDE integration are fully coupled [2,10,17,18,20,26]. We have successfully applied this "intermediate" approach in [5,6].…”

“…These schemes are "intermediate" between the static regridding methods [ 14,[22][23][24], where nodes remain fixed for intervals of time, and continuously moving grid methods, where the node movement and the PDE integration are fully coupled [2,10,17,18,20,26]. While the research in [5,6] has enabled us to identify a promising scheme, the implementation considered in those papers used fixed time-steps and did not allow a dynamic variation of the number of spatial grid-points.…”