An unconditionally stable second-order accurate ALE-FEM scheme for two-dimensional convection-diffusion problems (Q2902195)

For each \(t\in [0,T]\) let \(\Omega_t\) be a domain in \(\mathbb{R}^2\), where \(\Omega_0\) denotes the initial physical domain or a computational configuration \(\Omega_c\).NEWLINENEWLINE The authors consider the following convection-diffusion problem: \(u_t+\nabla\cdot(au)- \mu\Delta u= f\), \((x,t)\in Q_T\), \(u= 0\) for \(x\in \partial\Omega_t\), \(u(x,0)= u_0\), \(x\in\Omega_0\), \(\nabla\cdot a= 0\), where \(Q_T= \{(x,t): x\in\Omega_t, ~ t\in (0,T)\}\), \(a\) is a convective velocity term, \(\mu> 0\) is a constant coefficient of diffusivity.NEWLINENEWLINE Using an arbitrary Lagrangian-Euler (ALE) transformation from the domain \(\Omega_0\) onto the moving domain \(\Omega_t\), the authors reformulate the above problem. (Not an additional convection-like term appears.) Next, a weak form of the ALE formulation is obtained. Then finite element semi-discretization and finally fully discretization is formulated. Modifying the standard backward Euler scheme, the authors obtain a first-order method and modifying the Crank-Nicolson (CN) method, they obtain a second-order method. Both methods satisfy a discrete version of the geometric conservation law, i.e., they reproduce a constant solution. It is proved that the CN method is unconditionally stable. Numerical experiments are reported in details.