(Penalization error) Assume that

1. the coefficients of satisfy the definition ( Elliptic differential operator ),

2. the condition ( Assumption of coercivity 1 ) holds,

3. ,

4. ,

5. on

then for the solution of the problem ( Stationary penalized problem ) and the solution of the problem ( Stationary variational inequality problem ) the following estimate holds

We set in the problem ( Stationary penalized problem ):

and note that thus we obtain or We estimate the RHS using the propositions ( Energy estimates for the bilinear form B ) and ( Cauchy inequality ) and we estimate the LHS using the condition ( Assumption of coercivity 1 ). We obtain Using such estimate we revisit , use the same tools again and obtain

We introduce the quantity Thus it remains to show that

Note that . Indeed, also, since and on We set in the problem ( Stationary penalized problem ) and set in the problem ( Stationary variational inequality problem ): We add and and obtain Hence But hence Thus and by the condition ( Assumption of coercivity 1 ) and the propositions ( Energy estimates for the bilinear form B )-1, Together with this implies the statement of the proposition.