Fourth- and Higher-Order Finite Element Methods for the Incompressible Navier–Stokes Equations With Dirichlet Boundary Conditions

Abstract

Inspired by the unconstrained pressure Poisson equation (PPE) formulation [Liu, Liu, & Pego, Comm. Pure Appl. Math. 60 (2007): 1443–1487], we previously proposed the generic projection and unconstrained PPE (GePUP) formulation [Zhang, J. Sci. Comput. 67 (2016): 1134–1180] for numerically solving the incompressible Navier–Stokes equations (INSE) with no-slip boundary conditions. In GePUP, the main evolutionary variable does not have to be solenoidal, with its divergence controlled by a heat equation. This work presents GePUP–FEM, high-order finite-element solvers for the INSE under the framework of the method-of-lines. Continuous Lagrange finite elements of equal order are utilized for the velocity and pressure finite element spaces to discretize the weak form of GePUP in space, while high-order implicit–explicit Runge–Kutta methods are then employed to treat the stiff diffusion term implicitly and the other terms explicitly. Due to the implicit treatment of the diffusion term, the time step size is only restricted by convection. The solver is efficient in that advancing the solution at each time step only involves solving a sequence of linear systems either on the velocity or on the pressure, with geometric multigrid methods. Furthermore, the solver is enhanced with adaptive mesh refinement (AMR) so that the multiple length scales and time scales in flows at moderate or high Reynolds numbers can be efficiently resolved. Numerical tests with various Reynolds numbers are performed for the single-vortex test, the lid-driven cavity, and the flow past a cylinder/sphere, demonstrating the high-order accuracy of GePUP–FEM both in time and in space and its capability of accurately and efficiently resolving the right physics. Moreover, GePUP–FEM offers the flexibility in choosing velocity and pressure finite element spaces and is free of the standard inf-sup condition.