Functional A Posteriori Error Control for Conforming Mixed Approximations
of Coercive Problems with Lower Order Terms
Authors: Anjam Immanuel , Pauly Dirk 
Published: May 2016

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
*

⁢
A
⁢
x

+
x

=
f

${\mathrm{A}^{*}\mathrm{A}x+x=f}$

or in mixed formulation

A
*

⁢
y

+
x

=
f

${\mathrm{A}^{*}y+x=f}$

,

A
⁢
x

=
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
~

-

A
⁢

x
~

|

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
*

⁢
A
⁢
x

+

i
⁢
x

=
f

${\mathrm{A}^{*}\mathrm{A}x+ix=f}$

or in mixed formulation

A
*

⁢
y

+

i
⁢
x

=
f

${\mathrm{A}^{*}y+ix=f}$

,

A
⁢
x

=
y

${\mathrm{A}x=y}$

,
where i is the imaginary unit. For this problem we have the two-sided estimate

2

2

+
1

⁢

ℳ
i

⁢

(

x
~

,

y
~

)

≤

|

x
-

x
~

|

2

+

|

A
⁢

(

x
-

x
~

)

|

2

+

|

y
-

y
~

|

2

+

|

A
*

⁢

(

y
-

y
~

)

|

2

≤

2

2

-
1

⁢

ℳ
i

⁢

(

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

ℳ
i

⁢

(

x
~

,

y
~

)

:=

|

f
-

i
⁢

x
~

-

A
*

⁢

y
~

|

2

+

|

y
~

-

A
⁢

x
~

|

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.