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The error associated with a numerical solution is intimately related with the residual, that is the lack of verification of the equation by the approximated solution. The residual is computable but obtaining the exact error from the residual is as difficult as computing the exact solution. Residual type estimators provide error assessment tools based on post processing the residual. This post process is either explicit (integrating the residual) or implicit (solving local problems with the residual as source term). Some of the residual type estimates are guaranteed error bounds. The standard estimators aim at assessing the energy norm of the error. Goaloriented assessment is carried out by considering an auxiliary problem associated with the selected quantity of interest (the adjoint or dual problem). Thus, an error representation allows estimating the error in the quantity of interest as a post-process of the energy measures of the errors in both the original problem and the adjoint one.
CitacióChamoin, L., Diez, P. Preface. A: "Verifying Calculations: forty years on". Springer, 2016, p. V-VI.