Neumann boundary conditions

The optimization variable $\,z\,$ in (4.1) is taken as the vector

$$z:=(\,(y_{ij})_{\,(i,j) \in I(\bar{\Omega})}\,, \,(u_{ij})_{\,(i,j) \in I(\Gamma)}\,) \,\in I\!\! R^{\,N^2+4N+4N} \,.$$

Equality constraints are obtained by applying the five-point-star to the elliptic equation $\, - \Delta y(x) + d(x,y(x)) = 0 \,$ in (2.2) in all points $\,x_{ij}\,$ with $\,(i,j) \in I(\Omega)$:

$$\hspace*{-5mm} G^h_{ij}(z):= 4y_{ij} - y_{i+1,j} - y_{i-1,j} - y_{i,j+1} - y_{i,j-1} + h^2\,d(x_{ij},y_{ij}) = 0 \,.$$ (4.3)

The derivative $\,\partial_{\nu} y(x_{ij}) \,$ in the direction of the outward normal is approximated by the expression $\, y^{\,\nu}_{ij}/h\,$ where

$$y^{\,\nu}_{ij} := \; \left \{ \begin{array}{llll} \ y_{i0} ... ..., & \mbox{for} & j=N+1, & i=1,...,N \end{array}\right \} \, .$$ (4.4)

Then the discrete form of the Neumann boundary condition in (2.2) leads to the equality constraints

$$B^h(z):= y^{\nu}_{ij} - h\,b(x_{ij},y_{ij},u_{ij}) = 0\quad \mbox{for} \quad (i,j) \in I(\Gamma)\, . \quad$$ (4.5)

The control and state inequality constraints (2.3) and (2.4) yield the inequality constraints

$$S(x_{ij},y_{ij}) \leq 0 \,, \hspace*{12mm} (i,j) \in I(\bar{\Omega})\,,$$ (4.6)

$$C(x_{ij},y_{ij},u_{ij}) \leq 0 \,, \quad (i,j) \in I(\Gamma)\,.$$ (4.7)

Observe that these inequality constraints do not depend on the meshsize $\,h\,$. Later on, this fact will require a scaling of the Lagrange multipliers. The discretized form of the cost function (2.1) is

$$F^h(z):= h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij}) \,+\, h \sum_{(i,j) \in I(\Gamma)} g(x_{ij},y_{ij},u_{ij}) \,.$$ (4.8)

Then the relations (4.3)-(4.8) define an NLP-problem of the form (4.1). The Lagrangian function for this NLP-problem becomes

$$\hspace*{-10mm} L(z,q,\mu,\lambda):= \hspace*{-4mm} h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij}) \,+\, h \sum_{(i,j) \in I(\Gamma)} g(x_{ij},y_{ij},u_{ij})$$ (4.9)

$$\hspace*{-6mm} +\,\sum_{(i,j)\in I(\Omega)} \,q_{ij} G^h_{ij}(z) + \,\sum_{(i,j)\in I(\bar{\Omega})}\,\mu_{ij} S(x_{ij},y_{ij})\,$$

$$\hspace*{-6mm} +\, \sum_{(i,j)\in I(\Gamma)} [\,q_{ij} B^h(z) + \lambda_{ij} C(x_{ij},y_{ij},u_{ij})\,] \,.$$

The Lagrange multipliers $\,q=(q_{ij})_{(i,j) \in I(\bar{\Omega})}\,$, $\,\mu=(\mu_{ij})_{(i,j) \in I(\bar{\Omega)}}\,$ resp.
$\,\lambda=(\lambda_{ij})_{(i,j) \in I(\Gamma)}\,$ are associated with the equality constraints (4.3) and (4.5), the inequality constraints (4.6), resp. the inequality constraints (4.7). The multipliers $\,\lambda\,$ and $\,\mu\,$ satisfy complementarity conditions corresponding to (2.10):

$$\begin{eqnarray*} \begin{array}{llll} \lambda_{ij} \geq 0 & \mbox{and} & \; \lam... ... & \mbox{for all} \quad (i,j) \in I(\bar{\Omega}) \,. \end{array}\end{eqnarray*}$$

The discussion of the necessary conditions of optimality

$$\,0 = L_z = (\,(L_{y_{ij}})_{(i,j)\in I(\bar{\Omega}) }\,, (L_{u_{ij}})_{(i,j)\in I(\Gamma)}\,)$$

will be performed for different combinations of indices $\,(i,j)\,$. For indices $\, (i,j) \in I(\Omega)\,$ we obtain the relations

$$\begin{eqnarray*} 0 = L_{y_{ij}} = && \hspace*{-4mm} 4q_{ij} - q_{i+1,j} - q_{i-... ...+ h^2 \,f_y(x_{ij},y_{ij})\,+\, \mu_{ij}S_y(x_{ij},y_{ij}) \,. \end{eqnarray*}$$

Hence, the Lagrange multipliers $\,q=(q_{ij})\,$ satisfy the five-point-star difference equations for the adjoint equation $\, -\Delta \bar{q} + \bar{q}\,d_y + f_y + \,S_y\,\bar{\mu} = 0\,$ in (2.7) if we make the following identification for the Borel measure $\,\bar{\mu}\,$. Let $\,sq(h^2)\,$ denote a square centered at $\,x_{ij}\,$ with area $h^2$. Then we have the approximation

$$\hspace*{20mm} \int_{sq(h^2)} \, d\bar{\mu} \; \sim \; \mu_{ij} \,.$$ (4.10)

Recall the decomposition (2.11) of the measure $\,\bar{\mu} = \bar{\nu} \cdot dx\,+\,\bar{\nu}_s \cdot \bar{\mu}_s\,$. If the singular part of the measure vanishes, i.e. $\,\bar{\nu}_s \cdot \bar{\mu}_s=0,$ then (4.10) yields the approximation for the density $\,\bar{\nu}$:

$$\hspace*{20mm} \bar{\nu}(x_{ij})\, \sim\ \mu_{ij}/h^2 \,.$$ (4.11)

In case that the measure $\, \bar{\mu} = \bar{\nu}_s \cdot \delta(x-x_{ij})\,$ is a delta distribution, we obtain from (4.10) the relation

$$\hspace*{20mm} \bar{\nu}_s \, \sim\ \mu_{ij} \,.$$ (4.12)

For indices $\,(i,j) \in I(\Gamma)\,$ on the boundary we get, e.g. for $\,j=0,\,i=1,...,N\,$:

$$\begin{eqnarray*} 0= L_{y_{i0}} = && \hspace*{-6mm} -q_{i1} + q_{i0} -q_{i0}hb_y... ...0}\,C_y(x_{i0},y_{i0},u_{i0}) + \mu_{i0}\,S_y(x_{i0},y_{i0}) \,. \end{eqnarray*}$$

This is just the discrete version of the Neumann boundary condition (2.8) if we identify

$$\hspace*{10mm} h\,\bar{\lambda}(x_{i0})\,\sim\,\lambda_{i0} , \quad \int_{s(h)} d\bar{\mu} \sim \,\mu_{i0} \,,$$ (4.13)

where $\,s(h)\,$ is a line segment on $\,\Gamma\,$ of length $h$ centered at $\,x_{i0}\,$. For the special decomposition $\,\bar{\mu} =\bar{\nu}\cdot dx\,$ this leads to the identifications

$$\hspace*{10mm} \bar{\lambda}(x_{i0}) \, \sim \lambda_{i0} / h \,, \quad \bar{\nu}(x_{i0}) \sim \,\mu_{i0}/h \,.$$ (4.14)

Similar relations hold for other indices $\,(i,j) \in I(\Gamma)\,$. Finally, necessary conditions with respect to the control variables $\,u_{ij}, (i,j)\in I(\Gamma)\,$ are determined, e.g., for indices $j=0,\,i=1,...,N\,,$ by

$$0= L_{u_{i0}} = h g_u(x_{i0},y_{i0},u_{i0}) - q_{i0}\,h b_u... ...,y_{i0},u_{i0}) + \lambda_{i0}\,C_u(x_{i0},y_{i0},u_{i0}) \,.$$

This is the discrete version of the optimality condition (2.9) for the control, if we use again the identification $\,h\,\bar{\lambda}(x_{i0}) \,\sim\, \lambda_{i0} \,$.