Abstract :
The two principal ways of relocating nodes in approximation theory and the numerical solution of partial differential equations (PDEs) are through equidistribution and direct minimization. The benefits and limitations of both approaches are briefly discussed together with implementations which include both functional representation and the solution of steady PDEs.
Although equidistribution must be combined with existing numerical schemes when used to adapt the solution of a PDE, direct minimization can be extended automatically to the fairly wide class of PDEs governed by variational principles. Several iteration methods are described for determining optimal grids and solutions in this way, one of which uses the technique of the Moving Finite Element method. Examples are given from different areas of grid adaptation including functional approximation, the solution of Poisson and advection equations, and the shallow water equations. The extension to time-dependent problems is discussed.
