Guaranteed and robust $L^2$-norm a posteriori error estimates for 1D linear advection problems

Alexandre Ern (1, 2) , Martin Vohralík (2, 1) , Mohammad Zakerzadeh (2, 1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the $L_2$-norm of the error with the dual graph norm of the residual is established. This dual norm is showed to be localizable over vertex-based patch subdomains of the computational domain under the condition of the orthogonality of the residual to the piecewise affine hat functions. We show that this condition is valid for some well-known numerical methods including continuous/discontinuous Petrov-Galerkin and discontinuous Galerkin methods. Consequently, a well-posed local problem on each patch is identified, which leads to a global conforming reconstruction of the discrete solution. We prove that this reconstruction provides a guaranteed upper bound on the $L_2$ error. Moreover, up to a constant, it also gives local lower bounds on the $L_2$ error, where the generic constant is proven to be independent of mesh-refinement, polynomial degree of the approximation, and the advective velocity. This leads to robustness of our estimates with respect to the advection as well as the polynomial degree. All the above properties are verified in a series of numerical experiments, additionally leading to asymptotic exactness. Motivated by these results, we finally propose a heuristic extension of our methodology to any space dimension, achieved by solving local least-squares problems on vertex-based patches. Though not anymore guaranteed, the resulting error indicator is numerically robust with respect to both advection velocity and polynomial degree, for a collection of two-dimensional test cases including discontinuous solutions.

Mots clés: Petrov-Galerkin method; Linear advection problem; Discontinuous Galerkin method; A posteriori error estimate; Local efficiency; Advection robustness; Polynomial-degree robustness

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Martin Vohralík, Mohammad Zakerzadeh. Guaranteed and robust $L^2$-norm a posteriori error estimates for 1D linear advection problems. ESAIM: Mathematical Modelling and Numerical Analysis, 2021, 55, pp.S447-S474. ⟨10.1051/m2an/2020041⟩. ⟨hal-02105418v2⟩