An adaptive enriched semi-Lagrangian finite element method for coupled flow-transport problems

An adaptive enriched semi-Lagrangian finite element method is proposed for the numerical solution of coupled flow-transport problems on unstructured triangular meshes. The new method combines the semi-Lagrangian scheme to deal with the convection terms, the finite element discretization to manage irregular geometries, a direct conjugate-gradient algorithm to solve the generalized Stokes problem, and an adaptive L2-projection using quadrature rules to improve the efficiency and accuracy of the proposed method. In this study, the gradient of the temperature is used as an error indicator for the adaptation of enrichments by increasing the number of quadrature points where it is needed without refining the computational mesh. Unlike other adaptive finite element methods for coupled flow-transport problems, linear systems in the proposed enriched semi-Lagrangian finite element method preserve the same structure and size at each refinement in the adaptation procedure. In addition, due to the Lagrangian treatment of convection terms in this approach, the standard Courant–Friedrichs–Lewy condition is relaxed and the time truncation errors are reduced in the diffusion–reaction part. We assess the performance of the proposed method for a convection–diffusion problem with a known analytical solution and for the benchmark problem of thermal flow past a circular cylinder. We also solve a heat transport problem in the Mediterranean Sea to illustrate the ability of the method to resolve complex flow features in irregular geometries. Comparisons to the conventional semi-Lagrangian finite element method are also carried out in the current work. The obtained numerical results demonstrate the potential of the proposed method to capture the main flow features and support our expectations for an accurate and highly efficient enriched semi-Lagrangian finite element method for coupled flow-transport problems. •Development of an enriched semi-Lagrangian finite element method.•An efficient multilevel adaptive method for incompressible Navier–Stokes equations.•Resolve sharp gradients without requiring very fine meshes or very small time steps.•Numerical assessment for a transport problem in the Mediterranea.