Moving Finite Elements. I

We present new and general 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. For clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes, with Burgers’ equation as our test equation; but the generalization to much more general spaces and equations (including even certain previous “moving vorticity blobs” of the first author and S. Doss for the Navier–Stokes equations) becomes clear. In this paper we present the theoretical and computational details of our scheme, along with computational trial runs for Burgers’ equation showing that the nodes do concentrate sharply and move with the shocks as desired. The conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a Newton’s method for our implicit stiff ODE solver and were thus limited to ver...