Adaptive Mesh Movement in 1D

In this chapter we discuss more formally the principles of adaptive mesh movement in 1D. The underlying mesh selection problem itself is quite simple to state: If one wishes to approximate a given function u(x) using its values at a finite number of mesh points, how should these points be chosen? The answer can usually be given as follows: one chooses a so-called mesh density function ρ(x), which in some way indicates the error in the numerical approximation, and the mesh points are then placed in such a way that distances between them are smaller in regions where ρ(x) is larger, and the distances are larger in regions where ρ(x) is smaller. For the one-dimensional case, adaptivity is predicated on what is called equidistribution, which is considered in some detail here. The argument for choosing the mesh density function ρ(x) will normally be motivated by the desire to minimize an error in interpolating a function or by solving a differential equation, although in special cases other arguments such as one based on scaling invariance are used.