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This paper presents an adaptive Discontinuous-Galerkin finite element method to solve linear advection–reaction equations. The method was developed for the prediction of hemolysis in prosthetic devices, where the flow may include recirculation zones. To ensure the well-posedness of the problem, we solve the transient form of the PDE. Time discretization is achieved with an implicit Euler scheme in order to obtain the steady-state solution rapidly. Linear interpolation functions are used to approximate the hemolysis field. The a posteriori estimation of the error is obtained by solving an advection–reaction equation for the error, which is approximated with quadratic interpolation functions. Then, elementary estimations of the error are computed using a semi-norm developed to achieve the best possible results, even in the presence of recirculation. Code verification – to assess the convergence and conservation properties of the method – is performed with the method of manufactured solutions and with a conservation test using a sharp gaussian function as a reaction term. Excellent results are obtained in both cases. Predictions of hemolysis in a dialysis cannula are presented as an engineering application of our method. [Copyright &y& Elsevier]