Adaptively Refined Meshes

Abstract

In the previous chapter the solution of linear equation systems was discussed. Often there are even optimal order methods, so that an equation system of n equations can be solved in O(n) operations. This is optimal in the sense that any method which produces an output of n numbers has at least O(n) complexity, even if the method is as simple as write the vector zero. Of course, one may be able to tune given methods or find new ones with a smaller constant in the complexity, but in general a lower limit is reached. Nevertheless, it may be desirable to solve a given PDE even faster. In this book we will present two ways to do this; we introduce parallel algorithms in chapter 5. This is a general approach to reducing the complexity by using more than just one processor. Another way to reduce the complexity is to change the discretisation and to use fewer degrees of freedom. This can be done for functions which are smooth or regular enough by a higher order or by a sparse grid method, see also chapter 2.5. In cases of non-smooth functions, adaptive mesh refinement can be used to concentrate the computational effort to regions where it is needed most. We will cover the topic of adaptive mesh refinement in this chapter.