AbstractThe 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
${\mathrm{A}^{*}\mathrm{A}x+x=f}$
or in mixed formulation
${\mathrm{A}^{*}y+x=f}$, ${\mathrm{A}x=y}$,
where the exact solution $(x,y)$ is in $D(\mathrm{A})\times D(\mathrm{A}^{*})$.
Here ${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential)
operator and ${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming
mixed approximations, i.e., we assume that the approximation
${(\tilde{x},\tilde{y})}$ belongs to ${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. 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$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-%
\tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(%
\tilde{x},\tilde{y}),$where
${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}%
\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$
contains only known data.
Our second main result is an error estimate for all
equations of the class
${\mathrm{A}^{*}\mathrm{A}x+ix=f}$
or in mixed formulation
${\mathrm{A}^{*}y+ix=f}$, ${\mathrm{A}x=y}$,
where i is the imaginary unit. For this problem we have the two-sided estimate$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-%
\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}%
\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{%
\sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$where ${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}%
\tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$
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.
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