On a posteriori error estimators: A one-dimensional nodal diffusion adaptivity case

The linearity of the local, nodal diffusion problem admits the separation of the nodal flux φ into a source response flux phi — solution of the full-source, zero-incoming partial currents problem—, and a current response flux ψ — solution of the source-free, full-incoming partial currents problem. The source and current response fluxes are expanded in hyperbolic and trigonometric functions, and the coefficients found by the minimization of a posteriori nodal error estimators, which control the necessity for increasing the expansion orders and provide a quantification of the absolute errors involved. The iteration processes tested utilized a full solution of all the unknowns and a cumulative solution of only the added-new terms. A benchmark problem representing a MTR in one-dimensional slab geometry with exact solution was tested, and absolute errors in the domain and image spaces of the diffusion operator are presented. The cumulative and full solutions showed <0.08% and <0.08% spurious reactions per neutron born, and ΔKeff/Keff <300 pcm and <1 pcm, respectively. Conclusions are drawn regarding the pertinence and relevance of the estimation of nodal errors as a design necessity in future developments.