A posteriori error identities and estimates of modelling errors

The paper discusses a posteriori estimation methods for mathematical models based on partial differential equations. The analysis is based on functional identities of a special kind. They reflect the most general relations that hold for deviations from the exact solution of a boundary value problem. The identities do not use special properties of approximations and contain no mesh dependent constants. They are valid for any function in the admissible (energy) class. This universality makes it possible to control the accuracy of various numerical approximations and to compare solutions of mathematical models. These capabilities are demonstrated with the paradigm of modelling errors generated by simplifications of the original problem. The paper discusses three groups of problems, where errors of simplification have different origins. In the first case, the error arises as a result of simplifying coefficients of the equation. Errors of the second type are associated with simplification of geometry. Dimension reduction errors, that is reduction of dimension of the physical domain, form the third group, where the changes affect not only the geometry but also the overall dimensionality of the problem. It is shown that in any of these cases, the desired error estimates follow from the general a posteriori identity after proper specification of the functional spaces and operators associated with the boundary value problem. © 2024