This paper deals with the linear parabolic integro-differential problem \[ u_t(\mathbf{x},t)-\nabla\cdot(A\nabla u(\mathbf{x},t))-\int\limits_{0}^{t} \nabla\cdot(B\nabla u(\mathbf{x},\tau))d\tau=f(\mathbf{x},t), \; (\mathbf{x},t)\in\Omega\times(0,T], \]
\[ u=g(\mathbf{x},t), \; (\mathbf{x},t)\in\partial\Omega\times(0,T],\quad u(\mathbf{x},0)=\psi(\mathbf{x}), \; \mathbf{x}\in\Omega, \] where \(\mathbf{x}=(x_1,x_2),\;\Omega\subset \mathbb{R}^2\) is a bounded convex polygonal domain with the boundary \(\partial\Omega\), \(A=[a_{ij}(\mathbf{x},t)]_{2\times 2}\) and \(B=[b_{ij}(\mathbf{x},t)]_{2\times 2}\) are matrix-valued functions defined on \(\Omega\times (0,T]\).
The authors propose the semidiscrete and fully discrete weak Galerkin finite element schemes for this problem. Optimal error estimates are established for the corresponding numerical approximations in both \(L^2\) and \(H^1\) norms. Several illustrative numerical examples are presented in conclusion.