A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods
Provides rigorous error estimation for discontinuous Galerkin methods in convection-diffusion-reaction problems, offering explicit constants and robustness guarantees, which is incremental for the numerical analysis community.
The paper proposes two a posteriori error estimators for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods, proving reliability with explicit constants and robustness for dominant convection or reaction terms, with asymptotic exactness under superconvergence. Numerical tests confirm reliability and efficiency.
We consider some (anisotropic and piecewise constant) convection-diffusion-reaction problems in domains of R2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose two a posteriori error estimators based on gradient recovery by averaging. It is shown that these estimators give rise to an upper bound where the constant is explicitly known up to some additional terms that guarantees reliability. The lower bound is also established, one being robust when the convection term (or the reaction term) becomes dominant. Moreover, the estimator is asymptotically exact when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimators are confirmed by some numerical tests.