Advanced forms of functional a posteriori error estimates for elliptic problems.

Functional a posteriori estimates have been derived for elliptic and linear parabolic problems in [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals, 2000. S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, 2003. S. Repin, A posteriori error estimates for partial differential equations, 2008. S. Repin, A posteriori error estimation for nonlinear variational problems by duality theory, 1997] and some other publications. They provide computable upper bounds of the difference between the exact solution u and any approximation v lying in the admissible (energy) class. This paper is concerned with advanced forms of these estimates, which are discussed within the paradigm of the reaction-diffusion problem. In the first part of the paper, we derive guaranteed and computable upper bounds for problems which admit the decomposition of the domain into a set of simple (e.g., simplicial or polyhedral) subdomains. For this case an upper bound which involves constants in the Poincaré inequality for the corresponding subdomains is obtained. Estimates of this type can be helpful if computations are performed by the domain decomposition method. The second part of the paper is devoted to the derivation of two-sided error bounds in terms of weighted norms. Our analysis shows that the approach based on certain transformations of the basic integral identity earlier developed for energy error norms (see [S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, 2003. S. Repin, A posteriori error estimates for partial differential equations, 2008]) can be successfully applied to error estimation in weighted norms. [ABSTRACT FROM AUTHOR]