ABSTRACT
This chapter considers basic concepts of the convergence of a computed finite element solution, rates of convergence, a priori and a posteriori error estimations, computations of error in the finite element solutions, adaptive and self-adaptive finite element processes. There are three main sources of error in the finite element computational processes that are based on variationally consistent integral forms: due to approximation of boundaries, due to inadequate precision in computation and faulty or inadequate algorithms, and due to local approximations of the theoretical solution. In finite element computation processes as more degrees of freedom are added to a discretization the accuracy of the computed solution improves when the theoretical solution of the BVP is analytic or regular and when the integral forms are VC. Convergence of a finite element solution implies behavior of the error in the finite element solution as a function of the degrees of freedom or the characteristic length of the discretization.
