Interplay between discretization and algebraic computation in adaptive numerical solutionof elliptic pde problems

The Adaptive Finite Element Method (AFEM) for approximating solutions of PDE bound- ary value and eigenvalue problems is a numerical scheme that automatically and iteratively adapts the finite element space until a sufficiently accurate approximate solution is found. The adaptation process is based on a posteriori error estimators, and at each step of this process an algebraic problem (linear or nonlinear algebraic system or eigenvalue problem) has to be solved. In practical computations the solution of the algebraic problem cannot be obtained exactly. As a consequence, the algebraic error should be incorporated in the context of the AFEM and its a posteriori error estimators. The goal of this paper is to survey some exist- ing approaches in the AFEM context that consider the interplay between the finite element discretization and the algebraic computation. We believe that a better understanding of this interplay is of great importance for the future development in the area of numerically solving large-scale real-world motivated problems.