## 3 Extended Finite Element Method: Review
### Example: Poisson Equation  
We consider a Poisson problem and deliberately phrase the problem in a mathematically concrete and abstract fashion as we wish our domain specific language, which we will use to define the input to the form compiler, to inherit this syntax. The Poisson problem on \(\Omega\), with the boundary \(\Gamma=\partial\Omega\) partitioned such that \(\overline{\Gamma_{g}\cup\Gamma_{h}}=\Gamma\) and \(\Gamma_{g}\cap\Gamma_{h}=\emptyset\), involves finding \(u\) such that:  
\[-\Delta u =f \text{in}\ \Omega, \tag{9}\] \[u =0 \text{on}\ \Gamma_{g},\] (10) \[\
abla u\cdot\mathbf{n} =h \text{on}\ \Gamma_{h},\] (11) \[\
abla u^{+}\cdot\mathbf{n} =q \text{on}\ \Gamma_{d},\] (12) \[\llbracket\
abla u\rrbracket\cdot\mathbf{n} =0 \text{on}\ \Gamma_{d}, \tag{13}\]  
where \(f:\Omega\to\mathbb{R}\) a sufficiently regular source term, \(g:\Gamma_{g}\to\mathbb{R}\) and \(h:\Gamma_{h}\to\mathbb{R}\) are the prescribed boundary conditions, \(q\) is the flux across the discontinuity surface. The flux may be prescribed or it may be determined via a constitutive model. The jump operator is defined as \(\llbracket\
abla a\rrbracket=\
abla a^{+}-\
abla a^{-}\). Equation (13) therefore implies continuity of the flux across the surface \(\Gamma_{d}\). Assuming that the flux on the discontinuity surface is given by \(q=q\left(\llbracket u\rrbracket\right)\), the variational formulation of this problem reads: find \(u\in V\) such that  
\[\int_{\Omega\backslash\Gamma_{d}}\
abla v\cdot\
abla u\,d\Omega+\int_{\Gamma_ {d}}\llbracket v\rrbracket\,q\left(\llbracket u\rrbracket\right)\,d\Gamma=\int_ {\Omega}vf\,d\Omega,+\int_{\Gamma_{h}}vh\,d\Gamma\quad\forall v\in V, \tag{14}\]  
where  
\[V=\left\{v\in L^{2}\left(\Omega\right)\cap H^{1}\left(\Omega\backslash\Gamma_ {d}\right):v=0\ \text{on}\ \Gamma_{g}\right\}. \tag{15}\]  
We wish to construct a finite-dimensional counterpart of this problem using finite element basis functions without requiring that the triangulation of \(\Omega\) conform in any way to the discontinuity surface. We can do this by decomposing the finite element solution \(u_{h}\) along the same lines as Equation (5):  
\[u_{h}=\bar{u}_{h}+H_{d}\hat{u}_{h}. \tag{16}\]  
Decomposing the weight function \(v\) similarly, the variational problem can be expressed equivalently as: find \(\bar{u}_{h}\in\bar{V}\) and \(\hat{u}_{h}\in\hat{V}\) such that  
\[\int_{\Omega}\
abla\bar{v}\cdot\
abla\bar{u}\,d\Omega+\int_{ \Omega^{+}}\
abla\bar{v}\cdot\
abla\hat{u}\,d\Omega+\int_{\Omega^{+}}\
abla \hat{v}\cdot\
abla\left(\bar{u}+\hat{u}\right)\,d\Omega+\int_{\Gamma_{d}} \hat{v}q\left(\hat{u}\right)\,d\Gamma\\ =\int_{\Omega}\bar{v}f\,d\Omega+\int_{\Omega^{+}}\hat{v}f\,d \Omega+\int_{\Gamma_{h}}\bar{v}h\,d\Gamma+\int_{\Gamma_{h}^{+}}\hat{v}h\,d \Gamma\quad\forall\bar{v}\in\bar{V},\ \forall\hat{v}\in\hat{V}, \tag{17}\]  
where \(\Gamma_{h}^{+}=\Gamma_{h}\cap\partial\Omega^{+}\). The task now is to generate a suitable finite element space. Following from the expression for \(u_{h}\) in Equation (8), we define the finite element spaces  
\[\bar{V} =\left\{\bar{u}\in H^{1}\left(\Omega\right),\bar{u}|_{\Omega_{e}} \in P^{k_{1}}\left(\Omega_{e}\right)\forall e:u=0\ \text{on}\ \Gamma_{g}\right\}, \tag{18}\] \[u =0 \text{on}\ \Gamma_{g},\]0  
where \(\Omega_{d}\subset\Omega\) is a \"small\" region around the discontinuity surface \(\Gamma_{d}\). More precisely, \(\Omega_{d}\) as the union of the supports of all basis functions whose support is intersected by the discontinuity surface. The extended finite element formulation for the Poisson equation is now complete. In the definition of the finite element spaces, we have deliberately permitted the use of different order functions for \(\bar{u}_{h}\) and \(\hat{u}_{h}\) since it will be straightforward using the form compiler to generate code for \(k_{1}\
eq k_{2}\). For the examples  
are limited in terms of possible representations.