Numerical methods for evolutionary problems considering advection dominated and control

We study a time–dependent advection–diffusion equation with spatially varying advection and heterogeneous diffusivity under homogeneous Dirichlet conditions. The strong and weak formulations are derived and discretized by a conforming Galerkin finite element method, leading to the standard semi–discrete system with mass, stiffness, and advection matrices. Temporal integration is performed with an unconditionally stable implicit Euler scheme. A practical 2D assembly procedure based on a 7–point Gaussian quadrature is detailed. To assess discretization accuracy and mesh independence, we employ $L^{1}$, $L^{2}$, and $H^{1}$ norms together with the Grid Convergence Index (GCI), including Richardson extrapolation and an asymptotic range check via the convergence ratio. Beyond baseline simulations with elementwise constant advection, we formulate and solve a convex optimization problem for an advection field $\gamma_{\mathrm{opt}}$ that minimizes a quadratic functional and steers the solution within a prescribed subdomain. Numerical experiments on structured meshes (n=9,18,36 per direction) demonstrate consistent convergence, CAR values near unity, and reduced dispersion when using $\gamma_{\mathrm{opt}}$, while quantifying uncertainty through GCI. The results confirm the robustness and effectiveness of the proposed FEM framework for evolutionary advection–diffusion problems and provide a reproducible pathway for accuracy verification and transport-field design.