Numerical Analysis of the Finite Element Method

In this chapter, the authors examine some aspects of finite element convergence analysis. They introduce the mesh refinement technique that improves the quality of the calculated approximate solution. One can implement mesh refinement techniques from a priori information, knowledge of the singularity of the solution, for example, from a posteriori information from the calculated solution, obtained by error estimators. In the error analysis, situations that are more complicated but which are often encountered in practice were excluded, in order to simplify things: the non-polyhedral open subset (i.e. curvilinear polyhedra) case and the data approximation case. Finally, the authors discuss approximation methods involving approximate calculations of the data and/or of the (bi)linear forms – for example, via numerical quadrature formulas – as well as external, or non-conforming, methods, for which the approximation spaces are not included within the solution and/or test-function spaces.