CAM Seminar - 02/21/19

Abstract:  An efficient and accurate finite-element algorithm is described for the numerical solution of the incompressible Navier-Stokes (INS) equations. The algorithm based on a split-step scheme in conjunction with the standard finite-element method for spatial discretization solves the INS equations in an equivalent velocity-pressure formulation. The split-step scheme completely separates the pressure updates from the solution of velocity variables. When the pressure equation is formed explicitly, our algorithm avoids solving a saddle-point problem; therefore, its has more flexibility in choosing finite-element spaces for spatial discretization than the traditional mixed finite-element approach. Lagrange (piecewise-polynomial) elements of equal order for both velocity and pressure are used for efficiency. Accurate numerical boundary condition for the pressure equation is also investigated. Motivated by a post-processing technique that calculates derivatives of a finite element solution with super-convergent error estimates, an alternative numerical boundary condition is proposed for the pressure equation at the discrete level. The new numerical pressure boundary condition that can be regarded as a better implementation of the compatibility boundary condition improves the boundary-layer errors of the pressure solution. Analysis and numerical tests confirm that our scheme achieves high-order accuracy up-to the boundary.