## 3 Error Analysis for Regularized Delta and Heaviside Functions
### Truncation Error  
Considering that the support of a non-compact support function will be truncated at a certain distance from the interface \(\Gamma\), truncation will introduce a further source of approximation. However, it can be shown that the analytical error decreases for decreasing mesh size \(h\) while this does not generally happen in the case of compact support functions. However, the analytical error mainly depends on the truncation error, which is introduced in neglecting the contributions of quadrature points placed beyond a certain distance [22]. Hence we can fix \(\rho\) independently of the mesh size and increase the support until the numerical error is below an acceptable value.  
To recapitulate, we have two possibilities:  
1. we choose weight functions with compact support equal to the finite element size multiplied by a suitable integer number \(m\) that depends on \(h\), namely \(\ell_{\rho}\propto m\,h\)[1, 3]. This means that whenever the mesh changes, the weight function has to be changed accordingly, otherwise non-vanishing errors for decreasing mesh size are obtained.  
2. we choose _non-compact support_ weight functions whose Fourier transform \(\hat{\varphi}\) satisfies differentiability conditions; the length \(\ell_{\rho}\) will denote the truncated support of \(\varphi\). This makes it possible to have \(\ell_{\rho}\) independent of \(h\) when \(\ell_{\rho}\geq\gamma_{\hat{\varphi}}\,h\), where \(\gamma_{\hat{\varphi}}\) is the width of the truncated support of the Fourier transform \(\hat{\varphi}\).  
We want to investigate option 2. This requires evaluating the error due to truncation of the weight function. Denote the truncated delta function \(\delta_{\rho}^{\omega}\) as  
\[\delta_{\rho}^{\omega}(t)=\begin{cases}\delta_{\rho}(t)&\text{for }t\in[-\omega, \omega]\\ 0&\text{otherwise}\end{cases} \tag{20}\]  
where \(w=\ell_{\rho}/2\), \(\ell_{\rho}\) being the width of the truncated support of the weight function. [PERSON] and [PERSON] [PERSON] and [PERSON] (1970) showed that the analytical error \(\mathcal{E}_{\omega}\) associated with truncation can be neglected provided that  
\[\int_{-\omega}^{\omega}\delta_{\rho}^{\omega}(t)\,dt=1 \tag{21a}\] \[\int_{-\omega}^{\omega}\delta_{\rho}^{\omega}(t)\,t\,dt=0 \tag{21b}\]  
As \(\delta_{\rho}^{\omega}\) is even then  
\[\int_{-\infty}^{-\omega}\delta_{\rho}(t)\,t\,dt+\int_{\omega}^{\infty}\delta_ {\rho}(t)\,t\,dt=0 \tag{22}\]  
Consequently,  
\[\int_{-\infty}^{\infty}\delta_{\rho}(t)\,t\,dt=\int_{-\infty}^{-\omega}\delta_ {\rho}(t)\,t\,dt+\int_{\omega}^{\infty}\delta_{\rho}(t)\,t\,dt+\int_{-\omega}^ {\omega}\delta_{\rho}^{\omega}(t)\,t\,dt=0 \tag{23}\]  
Therefore Equation (23) implies that the truncated weight function satisfies the moment condition (21b). In order to prove condition (21a), we estimate  
\[I_{1}+I_{2}=\int_{-\infty}^{-\omega}\delta_{\rho}(t)\,dt+\int_{\omega}^{\infty }\delta_{\rho}(t)\,dt \tag{24}\]  
For instance taking \(\varphi=e^{-\frac{|t|}{\rho}}\) leads to \(I_{1}=I_{2}=\rho\,e^{-\frac{\omega}{\rho}}\). Hence the analytical error decreases for increasing support width \(\ell_{\rho}=2\omega\). The maximum value of the error, equal to \(\rho\), is reached for \(\omega=0\). In the following we will omit \(\omega\) and use \(\delta_{\rho}\) to indicate the truncated regularized delta.