Multilevel Adaptive Lagrange-Galerkin Methods for Unsteady Incompressible Viscous Flows

Abstract

A highly efficient multilevel adaptive Lagrange-Galerkin finite element method for unsteady incompressible viscous flows is proposed in this work. The novel approach has several advantages including (i) the convective part is handled by the modified method of characteristics, (ii) the complex and irregular geometries are discretized using the quadratic finite elements, and (iii) for more accuracy and efficiency a multilevel adaptive \(\mathrm {L}^2\)-projection using quadrature rules is employed. An error indicator based on the gradient of the velocity field is used in the current study for the multilevel adaptation. Contrary to the h-adaptive, p-adaptive and hp-adaptive finite element methods for incompressible flows, the resulted linear system in our Lagrange-Galerkin finite element method keeps the same fixed structure and size at each refinement in the adaptation procedure. To evaluate the performance of the proposed approach, we solve a coupled Burgers problem with known analytical solution for errors quantification then, we solve an incompressible flow past two circular cylinders to illustrate the performance of the multilevel adaptive algorithm.