Estimates for the difference between exact and approximate solutions of parabolic equations on the basis of Poincaré inequalities for traces of functions on the boundary

Abstract

We study a method for the derivation of majorants for the distance between the exact solution of an initial–boundary value reaction–convection–diffusion problem of the parabolic type and an arbitrary function in the corresponding energy class. We obtain an estimate (for the deviation from the exact solution) of a new type with the use of a maximally broad set of admissible fluxes. In the definition of this set, the requirement of pointwise continuity of normal components of the dual variable (which was a necessary condition in earlier-obtained estimates) is replaced by the requirement of continuity in the weak (integral) sense. This result can be achieved with the use of the domain decomposition and special embedding inequalities for functions with zero mean on part of the boundary or for functions with the zero mean over the entire domain.