A posteriori error bounds for quasi-linear Dirichlet and Neumann problems in full divergent form

Second-order quasi-linear Dirichlet and Neumann problems in four-term divergent form on a simply connected domain with a Lipschitz-continuous boundary of finite length are considered. Derivatives and primitives of distributions on the boundary are defined in such a way that for sufficiently smooth boundary distributions, these derivatives and primitives coincide with derivatives and primitives with respect to arc length on the boundary. Using these concepts, conjugate problems, that is, a pair of one Dirichlet and one Neumann problem, the minima of the energies of which add to zero, are introduced. From the concept 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 a posteriori error bounds for the norm of the difference of the approximate and exact solutions of the problem. These a posteriori bounds consist of a constant times the sum of the energies of the approximate solutions of the conjugate Dirichlet and Neumann problems and are easily constructed numerically.