The step of mathematical error estimation has two main purposes: First and most straight-forward, the error in the total simulation result shall be estimated and - if possible - reduced. Second and more relevant, the error can be localized and assigned to separate shape functions. This implies that these functions are inappropriate for forming the solution of a given problem and in general have to be altered. In terms of functions with local support the regions of support are divided into subregions where new functions are generated, which are used for a subsequent discretization and solution of the problem.
The measures which are taken for measuring the error are so-called a-posteriori [5,6,7,8,9] error estimation methods. These methods are proven to work on a class of linear problems whereas their correctness cannot be proven in general. Despite this fact these methods are commonly used in order to have a heuristic method for refining and fitting shape functions.