4.20 Error analysis

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Let ℘ denote the right-hand side A(S) = ℘ and let Vh = {Si} the trial space.

Regardless of whether we use the Hu-Washizu principle or the Hellinger–

Reissner principle the FE solution Sh is characterized by the property

a(Sh, Si) = _℘, Si_ Si ∈ Vh (4.160)

where a(., .) represents a symmetric bilinear form and _., ._ a linear form. To

make statements about the existence and uniqueness of an FE solution we

would need to see that these forms are continuous and coercive. We simply

assume that this is the case.

Most of the variational properties which we are used to attribute to FE

solutions, see Sect. 7.12, p. 568, can be carried over to mixed problems, because

to a large extent they are simply based on Green’s first identity and some

simple algebra. Hence it is evident that also the FE solution of mixed problems

satisfies the Galerkin orthogonality

a(S −Sh, Si) = 0 Si ∈ Vh , (4.161)

and also Tottenham’s equation holds for the solution of mixed problems, i.e.,

uh(x) =

_ l

0

[Nh

0 ε+ + εh0

N+ + Gh0

p ] dx . (• •) (4.162)

The proof is done as in the case of the original equation (1.210) on p. 64.

Recall (1.228) on p. 69, where we stated that

uh(x) = ph(G0) = ph(Gh0

)

_ _ _

•

= p(Gh0

)

_ _ _

• •

= (δ0, uh) = (δh

0 , uh) = (δh

0 , u)

(4.163)

and therefore we have for example as well

uh(x) =

_ l

0

[Nh

0 ε+

h + εh0

N+

h + Gh0

ph ]dx . ( • ) (4.164)

Of course also the basic formula for goal-oriented recovery techniques holds

as well

|e(x)| = |a(S0 −Sh

0 ,S −Sh)| ≤ ||S0 −Sh

0

||E ||S − Sh||E (4.165)

where ||S||2

E = a(S, S).

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