A posteriori identities for measures of deviation from exact solutions of nonlinear boundary value problems

Functional identities that hold for deviations from the exact solutions of boundary value and initial boundary value problems with monotone operators are obtained. These identities hold for any function from the corresponding functional class that contains the exact solution of the problem. The left-hand side of an identity is the sum of terms that measure deviation of the approximate solution from the exact one. It is shown that these measures are natural characteristics of the accuracy of approximate solutions. In some cases, the right-hand side of the identity contains only known data of the problem and functions that characterize the approximate solution. Such an identity can be directly used for error control. In other cases, the right-hand side includes unknown functions. However, they can be eliminated to obtain fully computable two-sided bounds. In this case, it is necessary to use special functional inequalities relating the deviation measures to the properties of the monotone operator under consideration. As an example, such bounds and the exact values of the corresponding constants are obtained for a class of problems with the $\alpha$-Laplacian operator. It is shown that the identities and the resulting bounds make it possible to estimate the error of any approximation regardless of the method used to obtain it. In addition, they open a way for comparing exact solutions of problems with different data, which makes it possible to evaluate the errors of mathematical models, e.g., those that arise when the coefficients of a differential equation are simplified. In the first part of the paper, the theory and applications concern stationary models, and then the main results are extended to evolutionary models with monotone spatial operators.