Section: New Results

A posteriori analysis and computational optimization

In 2015, E. Creusé et al. have developed a posteriori error estimators for the harmonic potential formulations of the Maxwell system, in order to simulate eddy-current problems arising in the context of quasi-static approximations. The originality of our contribution is to provide estimators with sharp bounds and explicit constants. It was achieved by solving in the same time the so-called "$𝐀/\varphi$" and "$𝐓/\Omega$" potential formulations [38] . If this way to proceed was already known and usually used for stationary problems, the extension to harmonic ones constitutes the novelty of our contribution. It was in particular necessary to prove some superconvergence properties of additional terms. The reliability as well as the local efficiency of the derived estimator have been established without any generic constant, and numerical tests clearly illustrate their optimal behavior, from academic benchmarks to more industrial ones.

Another track to optimize the computational effort consists in refining and coarsening the model. This approach is based on the following ansatz : the more the model is complex, the more expensive are the computations. This approach was used by F. Filbet and T. Rey in [23] to simulate kinetic equations, the kinetic equations being replaced by cheaper hydrodynamic limits when it is relevant. The same idea was used in H. Mathis et al. [27] in order to simulate complex flows modeled by hyperbolic systems with relaxation. A rigorous error analysis of such a model adaptation procedure was performed on a simplified model by C. Cancès et al. in [13] .