## Minimization of functional majorant in a posteriori error analysis based on $$H$$(div) multigrid-preconditioned CG method.(English)Zbl 1200.65095

Summary: We consider a Poisson boundary value problem and its functional a posteriori error estimate derived by S. Repin in 1999 [Probl. Mat. Anal. 17, 199–226 (1997; Zbl 0941.65059)]. The estimate majorizes the $$H^{1}$$ seminorm of the error of the discrete solution computed by FEM method and contains a free ux variable from the $$H$$(div) space. In order to keep the estimate sharp, a procedure for the minimization of the majorant term with respect to the ux variable is introduced, computing the free ux variable from a global linear system of equations. Since the linear system is symmetric and positive definite, few iterations of a conjugate gradient method with a geometrical multigrid preconditioner are applied. Numerical techniques are demonstrated on one benchmark example with a smooth solution on a unit square domain including the computation of the approximate value of the constant in Friedrichs’ inequality.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Zbl 0941.65059

mfem
Full Text:

### References:

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