An adaptive Lagrange-Galerkin method for the numerical solution of the Navier-Stokes equations

We shall examine the numerical solutions of two dimensional incompressible fluid flow problems via spectral methods. The focus will be on Galerkin methods with the aim of constructing a Lagrange-Galerkin method which is adaptive in time. A comparison will also be made between the Lagrange-Galerkin method in both its adaptive and unadaptive forms with the pseudospectral method. The pseudospectral method has been chosen since it is a widely accepted standard method for solving periodic fluid flow problems.;Since the Navier-Stokes equations, which govern the motion of incompressible fluids are nonlinear, there are often difficulties in computing the numerical solution. The main difficulty with the pseudospectral method is that it is only conditionally stable. This is the motivation for using the Lagrange-Galerkin method instead since it is unconditionally stable. So the only consideration that needs to be made for the step size is determined by how accurate the solution needs to be. The motivation for the adaptive Lagrange-Galerkin method is that it is a faster unconditionally stable method as opposed to the pseudospectral method.