Estimate of penalization error for stationary problem.

roposition

(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. $\psi\geq0$ on $\partial U$

then for the solution $u_{\varepsilon}$ of the problem ( Stationary penalized problem ) and the solution of the problem ( Stationary variational inequality problem ) the following estimate holds MATH

Proof

and note that MATH thus we obtain MATH or MATH 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 MATH Using such estimate we revisit , use the same tools again and obtain MATH

We introduce the quantity MATH Thus it remains to show that MATH

Note that . Indeed, MATH also, since and $\psi\geq0$ on $\partial U$ MATH We set $v=r_{\varepsilon}$ in the problem ( Stationary penalized problem ) MATH and set in the problem ( Stationary variational inequality problem ): MATH We add $\left( \&\right)$ and and obtain MATH Hence MATH But $u\in K$ hence MATH Thus MATH and by the condition ( Assumption of coercivity 1 ) and the propositions ( Energy estimates for the bilinear form B )-1, MATH Together with this implies the statement of the proposition.