# A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D
## 2 Model Problem and Weak Formulation  
Given a Lebesgue measurable set \(\Omega\), we denote by \(L^{p}(\Omega),\ 1\leq p\leq\infty\) the Lebesgue spaces. When \(p=2\), the space \(L^{2}(\Omega)\) is equipped with an inner product \(\langle\cdot,\cdot\rangle\). For an integer \(m>0\), we use the standard notation for Sobolev spaces \(W^{m,p}(\omega)\) with \(1\leq p\leq\infty\). When \(p=2\), we denote \(W^{m,2}(\Omega)\) by \(H^{m}(\Omega)\). The function space \(H^{1}_{0}(\Omega)\) consists of elements from \(H^{1}(\Omega)\) that vanish on the boundary of \(\Omega\), where the boundary values are to be interpreted in the sense of a trace.  
Consider the following initial-boundary value problem for a linear PIDE of the form  
\[\frac{\partial u}{\partial t}(\mathbf{x},t)+\mathcal{A}u(\mathbf{x},t) =\int_{0}^{t}\mathcal{B}(t,s)u(\mathbf{x},s)\mathrm{d}s+f(\mathbf{x},t), \quad(\mathbf{x},t)\in\Omega\times(0,T], \tag{1}\] \[u(\mathbf{x},t) =0,\quad(\mathbf{x},t)\in\partial\Omega\times[0,T],\] (2) \[u(\mathbf{x},0) =u_{0}(\mathbf{x}),\quad\mathbf{x}\in\Omega. \tag{3}\]  
Here, \(0<T<\infty\) and \(\Omega\subset\mathbb{R}^{2}\) is a bounded Lipschitz domain with boundary \(\partial\Omega\). Further, \(\mathcal{A}u(\mathbf{x},t)=-\Delta u(\mathbf{x},t)\) and  
\[\mathcal{B}(t,s)u(\mathbf{x},s)=-\
abla\cdot(B(t,s)\
abla u(\mathbf{x},s)), \tag{4}\]where \(\Delta\) denotes the Laplacian and \(\
abla\) denotes the spatial gradient. We assume that the coefficient matrix \(B(t,s)=\{b_{ij}(\mathbf{x};t,s)\}\) is \(2\times 2\) in \(L^{\infty}(\Omega)^{2\times 2}\). If we assume the initial function \(u_{0}(\mathbf{x})\) to be in \(H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\), the source term \(f(\mathbf{x},t)\) to be in \(L^{2}(0;T;L^{2}(\Omega))\) and  
\[\max_{\bar{\Omega}\times\{0\leq s\leq t\leq T\}}\big{|}\frac{\partial}{ \partial x}b_{i,j}(x;t,s)|<\infty,\]  
then the problem (1) admits a unique solution  
\[u\in L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega )).\]  
The existence results are discussed in detail in Chapter 2 of [9]. For regularity and stability results of such problems, please refer to [10; 28].  
Let \(a(\cdot,\cdot)\ :\ H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)\to\mathbb{R}\) be the bilinear form corresponding to the operator \(\mathcal{A}\) defined by  
\[a(\phi,\psi):=\langle\
abla\phi,\
abla\psi\rangle\ \ \ \ \ \forall\ \phi,\psi\in H^{1}_{0}(\Omega).\]  
Similarly, let \(b(t,s;\cdot,\cdot)\) be the bilinear form corresponding to the operator \(\mathcal{B}(t,s)\) defined on \(H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)\) by  
\[b(t,s;\phi(s),\psi):=\langle B(t,s)\
abla\phi(s),\
abla\psi\rangle\ \ \ \ \ \forall\ \phi(s),\psi\in H^{1}_{0}(\Omega).\]  
Then, the weak formulation of the problem (1) may be stated as follows: find \(u:[0,T]\to H^{1}_{0}(\Omega)\) such that, for all \(t\in(0,T]\),  
\[\int_{\Omega}\frac{\partial u}{\partial t}\phi\,\mathrm{d}\mathbf{x} ^{\prime}+a(u,\phi) =\int_{0}^{t}b(t,s;u(s),\phi)\,\mathrm{d}s+\int_{\Omega}f\phi\, \mathrm{d}\mathbf{x}^{\prime}\ \ \forall\phi\in H^{1}_{0}(\Omega), \tag{5}\] \[u(\cdot,0) =u_{0}.\]