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

This paper addresses the computation of guaranteed upper and lower bounds f or the energy norm of the exact error in the finite element solution, and t he exact error in any bounded linear functional. These bounds are construct ed by employing approximate solutions of the element residual problems with equilibrated residual loads. The one-dimensional setting is used for the c larity of the ideas. All the arguments employed can be extended to the high er-dimensional case which will be discussed in Part II of this paper. The m ain result presented here is that the computed bounds are guaranteed for th e exact error and not the error with respect to an enriched finite element solution, like the bounds proposed by other investigators and the bounds ar e guaranteed for any mesh, however coarse it may be. The quality of the bou nds can be controlled by employing an inexpensive iterative scheme. (C) 199 9 Elsevier Science S.A. All rights reserved.