Estimates of the deviation from exact solutions of boundary value problems in measures stronger than the energy norm

The paper is concerned with estimates of the difference between a given function and the exact solution of an elliptic boundary value problem. Estimates of this type have been derived earlier in terms of the natural energy norm. In this work, an approach is proposed to obtain stronger measures of the deviation and relevant estimates applicable if the exact solution and the approximation have additional regularity (with respect to the order of integrability). These measures include the standard energy norm as a simple special case. A general approach is proposed to construct various measures based on using an auxiliary variational problem. Two classes of measures whose properties are close to those of the ${L}^{q}$ and ${L}^{\infty}$ norms are studied in more detail. Their properties are established, and explicitly computable two-sided estimates (minorants and majorants) that involve only known functions are constructed.