A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain based a posteriori error estimate which yields self-equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux-free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds.

N. Parés, P. Díez, A. Huerta. Computable exact bounds for linear outputs from stabilized solutions of the advection-diffusion-reaction equation. Int. J. Numer. Meth. Engng 93(5) (2012) DOI 10.1002/nme.4396

V. Rey, P. Gosselet, C. Rey. Strict lower bounds with separation of sources of error in non-overlapping domain decomposition methods. Int. J. Numer. Meth. Engng 108(9) (2016) DOI 10.1002/nme.5244

A. Parret-Fréaud, V. Rey, P. Gosselet, C. Rey. Improved recovery of admissible stress in domain decomposition methods - application to heterogeneous structures and new error bounds for FETI-DP. Int. J. Numer. Meth. Engng 111(1) (2017) DOI 10.1002/nme.5462

P. Allier, L. Chamoin, P. Ladevèze. Towards simplified and optimized a posteriori error estimation using PGD reduced models. Int J Numer Methods Eng 113(6) (2017) DOI 10.1002/nme.5695

K. Olesen, B. Gervang, J. Reddy, M. Gerritsma. A higher-order equilibrium finite element method. Int J Numer Methods Eng 114(12) (2018) DOI 10.1002/nme.5785

N. Parés, P. Díez. A new 3D equilibrated residual method improving accuracy and efficiency of flux‐free error estimates. Int J Numer Methods Eng 120(4) (2019) DOI 10.1002/nme.6141

A. Ern, M. Vohralík. Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations. SIAM J. Numer. Anal. 53(2) DOI 10.1137/130950100

A. Allendes, F. Durán, R. Rankin. Error estimation for low-order adaptive finite element approximations for fluid flow problems. IMA J Numer Anal 36(4) (2015) DOI 10.1093/imanum/drv031

M. Ainsworth, R. Rankin. Computable error bounds for finite element approximation on nonpolygonal domains. IMA J Numer Anal (2016) DOI 10.1093/imanum/drw040

F. Pled, L. Chamoin, P. Ladevèze. On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples. Int. J. Numer. Meth. Engng. 88(5) (2011) DOI 10.1002/nme.3180

M. Ainsworth, R. Rankin. Guaranteed computable bounds on quantities of interest in finite element computations. Int. J. Numer. Meth. Engng 89(13) (2012) DOI 10.1002/nme.3276

F. Larsson. A Simple Anisotropic Mesh-Refinement Strategy for Triangular Elements in 2D. ISRN Applied Mathematics 2012 DOI 10.5402/2012/134097

L. Chamoin, P. Allier, B. Marchand. Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures. Adv. Model. and Simul. in Eng. Sci. 3(1) (2016) DOI 10.1186/s40323-016-0073-9

L. Steffens, N. Parés, P. Díez. Goal-oriented h-adaptivity for the Helmholtz equation: error estimates, local indicators and refinement strategies. Comput Mech 47(6) (2011) DOI 10.1007/s00466-010-0557-2

F. Pled, L. Chamoin, P. Ladevèze. An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress fields in finite element analyses. Comput Mech 49(3) (2011) DOI 10.1007/s00466-011-0645-y