Guaranteed computable bounds for the exact error in the finite element solution Part I: One-dimensional model problem Original Research Article

This paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution, and the exact error in any bounded linear functional. These bounds are constructed by employing approximate solutions of the element residual problems with equilibrated residual loads. The one-dimensional setting is used for the clarity of the ideas. All the arguments employed can be extended to the higher-dimensional case which will be discussed in Part II of this paper. The main result presented here is that the computed bounds are guaranteed for the exact error and not the error with respect to an enriched finite element solution, like the bounds proposed by other investigators and the bounds are guaranteed for any mesh, however coarse it may be. The quality of the bounds can be controlled by employing an inexpensive iterative scheme.