We are going to solve the following differential problem: given a sufficiently
regular function $g$ on $\Gamma$, find the solution $u$ to
-@f{eqnarray*}
-- \Delta u + \gamma^T \lambda &=& 0 & \text{ in } \Omega\\
-\gamma u &=& g & \text{ in } \Gamma \\
-u & = & 0 & \text{ on } \partial\Omega.
+@f{eqnarray*}{
+- \Delta u + \gamma^T \lambda &=& 0 \text{ in } \Omega\\
+\gamma u &=& g \text{ in } \Gamma \\
+u & = & 0 \text{ on } \partial\Omega.
@f}
This is a constrained problem, where we are looking for a harmonic function $u$
conditions on $\partial\Omega$, we obtain the following variational problem:
Given a sufficiently regular function $g$ on $\Gamma$, find the solution $u$ to
-@f{eqnarray*}
-(\nabla u, \nabla v)_{\Omega} + (\lambda, \gamma v)_{\Gamma} &=& 0 & \forall v \in V(\Omega) \\
-(\gamma u, q)_{\Gamma} &=& (g,q)_{\Gamma} & \forall q \in Q(\Gamma),
+@f{eqnarray*}{
+(\nabla u, \nabla v)_{\Omega} + (\lambda, \gamma v)_{\Gamma} &=& 0 \qquad \forall v \in V(\Omega) \\
+(\gamma u, q)_{\Gamma} &=& (g,q)_{\Gamma} \qquad \forall q \in Q(\Gamma),
@f}
where $(\cdot, \cdot)_{\Omega}$ and $(\cdot, \cdot)_{\Gamma}$ represent,
where
-@f{eqnarray*}
-K_{ij} &:=& (\nabla v_j, \nabla v_i)_\Omega & i,j=1,\dots,n \\
-C_{\alpha j} &:=& (v_j, q_\alpha)_\Gamma &j=1,\dots,n, \alpha = 1,\dots, m \\\\
-G_{\alpha} &:=& (g, q_\alpha)_\Gamma & \alpha = 1,\dots, m.
+@f{eqnarray*}{
+K_{ij} &:=& (\nabla v_j, \nabla v_i)_\Omega \qquad i,j=1,\dots,n \\
+C_{\alpha j} &:=& (v_j, q_\alpha)_\Gamma \qquad j=1,\dots,n, \alpha = 1,\dots, m \\\\
+G_{\alpha} &:=& (g, q_\alpha)_\Gamma \qquad \alpha = 1,\dots, m.
@f}
While the matrix $K$ is the standard stiffness matrix for the Poisson problem on
To evaluate $(v_j \circ F_{K}) (\hat x_i)$ the following steps needs to be
taken (as shown in the picture below):
-- For a given cell $K$ in $\Gamma$ compute the real point $y_i \coloneqq F_{K}
-(\hat x_i)$, where $x_i$ is one of the quadrature points used for the integral
-on $K \subseteq \Gamma$.
+- For a given cell $K$ in $\Gamma$ compute the real point $y_i := F_{K} (\hat
+x_i)$, where $x_i$ is one of the quadrature points used for the integral on $K
+\subseteq \Gamma$.
- Find the cell of $\Omega$ in which $y_i$ lies. We shall call this element $T$.
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