A posteriori error estimates for primal hybrid finite element methods


by Victor B. Oliari, Paulo Rafael Bosing, Denise de Siqueira, Philippe R.B. Devloo, presented at  XLII Ibero-Latin-American Congress on Computational Methods in Engineering (CILAMCE-2021) | 3rd Pan American Congress on Computational Mechanics, Novermber 2021.


We present new fully computable a posteriori error estimates for the primal hybrid finite element methods based on equilibrated flux and potential reconstructions. The reconstructed potential is obtained from a local L2 orthogonal projection of the gradient of the numerical solution, with a boundary continuous restriction that comes from a smoothing process applied to the trace of the numerical solution over the mesh skeleton. The equilibrated flux is the solution of a local mixed form problem with a Neumann boundary condition given by the Lagrange multiplier of the hybrid finite element method solution. To establish the a posteriori estimates we divide the error into conforming and non-conforming parts. For the former one, a slight modification of the a posteriori error estimate proposed by Vohral ́ık [1] is applied, whilst the latter is bounded by the difference of the gradient of the numerical solution and the reconstructed potential. Numerical results performed in the environment PZ Devloo [2], show the efficiency of this strategy when it is applied for some test model problems.