## 3 Error Analysis for Regularized Delta and Heaviside Functions  
Let \(\mathcal{F}(\mathbf{x})\) be a discontinuous or singular function, and consider the problem of the exact integration of the product \(\mathcal{F}(\mathbf{x})\,v(\mathbf{x})\) over the volume \(V\)  
\[\int_{V}\mathcal{F}(\mathbf{x})v(\mathbf{x})\,d\mathbf{x} \tag{8}\]  
where \(v(\mathbf{x})\) is a sufficiently regular scalar function for the Equation (8) to make sense [18, 23]. Assume that the function \(\mathcal{F}_{\rho}\) is the regularized version of \(\mathcal{F}\), namely \(\mathcal{F}_{\rho}\) is either equal to the regularized Heaviside function \(\mathcal{H}_{\rho}\) or the regularized Delta function \(\delta_{\rho}\). Our aim is to compute the integral Equation (8) through a quadrature procedure applied to its regularized version. Let  
\[\mathcal{E}_{\rho,v}(\mathcal{F}_{\rho})=\int_{V}\mathcal{F}(\mathbf{x})v( \mathbf{x})\,d\mathbf{x}-\int_{V}\mathcal{F}_{\rho}(\mathbf{x})v(\mathbf{x}) \,d\mathbf{x} \tag{9}\]  
be the _analytical error_ when replacing the Heaviside function by its regularized version and  
\[E_{\text{quad},v}(\mathcal{F}_{\rho})_{V}=\int_{V}\mathcal{F}_{\rho}(\mathbf{x })v(\mathbf{x})\,d\mathbf{x}-\text{quad}\bigg{(}\mathcal{F}_{\rho}(\mathbf{x })v(\mathbf{x})\bigg{)}_{V} \tag{10}\]  
be the _numerical error_ related to the numerical quadrature. [PERSON] [18] showed that the total error associated with the replacement of the exact integral Equation (8) with its approximated quadrature can be expressed as  
\[\mathcal{E}_{\text{tot},v}(\mathcal{F}_{\rho})=\int_{V}\mathcal{F}(\mathbf{x} )v(\mathbf{x})\,d\mathbf{x}-\text{quad}\bigg{(}\mathcal{F}_{\rho}(\mathbf{x} )v(\mathbf{x})\bigg{)}_{V}=\mathcal{E}_{\rho,v}(\mathcal{F}_{\rho})+ \mathcal{E}_{\text{quad},v}(\mathcal{F}_{\rho})_{V} \tag{11}\]  
Following [18, 25], analytical and numerical errors are competing source of error as  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**weight function** & \\ \hline hat function & \(\varphi_{H}(t)=1-\frac{|t|}{\rho}\) & if \(t\leq 1\) \(\quad,\varphi_{H}(t)=0\) & otherwise \\ bell & \(\varphi_{B}(t)=\bigg{(}1-\frac{t^{2}}{\rho^{2}}\bigg{)}^{4}\) & if \(|t|\leq\rho\), \(\quad\varphi_{B}(t)=0\) & otherwise \\ Laplacian & \(\varphi_{L}(t)=e^{\frac{-|t|}{\rho}}\) & \\ Gaussian & \(\varphi_{G}(t)=e^{-\frac{t^{2}}{\rho}}\) & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Examples of weight functions.  
* the analytical error \(E_{\rho,v}(\mathcal{F}_{\rho})\) typically increases with increasing \(\rho\) ;
* the numerical error \(E_{\text{quad},v}(\mathcal{F}_{\rho})_{V}\) is large for \(\rho\) small compared with the mesh size and decreases for increasing \(\rho\).  
More precisely, the \(q-\)order of the moment condition  
\[\int_{-\infty}^{\infty}\mathcal{F}_{\rho}(x)\,x^{r}\,dx=\begin{cases}1&\text{ if}\quad r=0\\ 0&\text{if}\quad 1\leq r\leq q\end{cases} \tag{12}\]  
influences the analytical error of the regularized delta approximation [18]. Moreover, while the analytical error depends on the number of satisfied moment conditions, differentiability influences the numerical error.  
In particular, consider an error analysis based on the progressive mesh refinement for fixed \(\rho\). If the mesh size \(h\) is rather large compared with the regularization parameter \(\rho\), the numerical error \(E_{\text{quad},v}(\mathcal{F}_{\rho})\) is expected to compete with the analytical error \(E_{\rho,v}(\mathcal{F}_{\rho})\). Subsequently, when \(h/\rho\) decreases, the analytical error will be more evident while the numerical error decreases.  
In the following sections we focus on the analytical and the numerical errors corresponding to regularizing the Delta function, namely \(\mathcal{F}_{\rho}=\delta_{\rho}\).