### Abstract

The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y , respectively. Our first main result is an error equality for all equations of the class A*Ax+x=f${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation A*y+x=f${\mathrm{A}^{*}y+x=f}$, Ax=y${\mathrm{A}x=y}$, where the exact solution (x,y)$(x,y)$ is in D(A)×D(A*)$D(\mathrm{A})\times D(\mathrm{A}^{*})$. Here A${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and A*${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation (x~,y~)${(\tilde{x},\tilde{y})}$ belongs to D(A)×D(A*)${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality |x-x~|2+|A(x-x~)|2+|y-y~|2+|A*(y-y~)|2=ℳ(x~,y~),$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$ where ℳ(x~,y~):=|f-x~-A*y~|2+|y~-Ax~|2${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class A*Ax+ix=f${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation A*y+ix=f${\mathrm{A}^{*}y+ix=f}$, Ax=y${\mathrm{A}x=y}$, where i is the imaginary unit. For this problem we have the two-sided estimate 22+1ℳi(x~,y~)≤|x-x~|2+|A(x-x~)|2+|y-y~|2+|A*(y-y~)|2≤22-1ℳi(x~,y~),$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$ where ℳi(x~,y~):=|f-ix~-A*y~|2+|y~-Ax~|2${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.