Abstract
A posteriori error analysis in numerical partial differential equations aims at providing sufficiently accurate information about the distance of the numerically computed approximation to the true solution. Besides estimating the total error, a posteriori analysis should also provide information about its discretization and (inexact) algebraic computation parts.
This issue has been addressed by many authors using different approaches. Historically, probably the first and practically very important approach is based on combination of the classical residual-based bound on the discretization error with the adaptive hierarchy of discretizations and computations that allow to incorporate, using various heuristic arguments, the algebraic error.
Motivated by some recent publications, this text uses a complementary approach and examines subtleties of the (generalized) residual-based a posteriori error estimator for the total error that rigorously accounts for the algebraic part of the error. The aim is to show on the standard Poisson model problem example, which is used here as a case study, that a rigorous incorporation of the algebraic error represents an intriguing problem that is not yet completely resolved.
That should be of concern in h-adaptivity approaches, where the refinement of the mesh is determined using the residual-based a posteriori error estimator assuming Galerkin orthogonality. The commonly used terminology such as 'guaranteed computable upper bounds' should be in the presence of algebraic error cautiously examined.