A posteriori error estimates for primal hybrid finite element methods

Resumo

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 L
2
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.