A posteriors error bounds for inhomogeneous linear dirichlet and neumann problems

Second-order inhomogeneous linear Dirichlet and Neumann problems in divergent form on a simply-connected to-dimensional domain with Lipschitz-continious boundary of finite length are considered. Conjugate problems, that is, a pair of one Dirichlet and one Neumann problem the minima of energies of which add to a known constant, are introduced. From the conceppt of conjugate problems, two-sided bounds for the energy of the exact solution of any given Dirichlet or Neumann problem are constructed. These two-sided bounds for the energy at the exact solution are in turn used to obtain easily calculable a posteriori error bounds for the norm of the difference of the approximate and exact solutions of the given problem