Estimated error bounds for finite element solutions of elliptic boundary value problems Original Research Article

A new error estimate is introduced which adds to the theory of error bounds. First, the statistically (SAS) and kinematically admissible fields (KAS), are studied. Then, the theory of error bounds using these fields is presented. Due to the inherent complexities involved in the determination of statically admissible stress field (SAS), a quasi-statically admissible stress (quasi-SAS) field is used instead, and a so-called error meaure, the estimated error bound, is defined. This error measure is determined by the sum of two terms, one of which is expressed by the normalized energy norm of the distance between the KAS field, computed by the displacement finite element method, to a quasi-SAS field. The other term is the error, in normalized energy norm, due to the application of quasi-SAS field instead of the statically admissible field. Because this term is also difficult to compute, an upper bound for this error is derived. In addition, some approximate expressions for this term are computed. The summation of the two terms comprises the estimated error bound. To show the effectiveness of this error measure, it is calculated for some test problems and compared with the exact errors in the solutions, when they exists.