An Efficient Least-Squares Finite Element Method for Incompressible Flows and Transport Processes

Simulations of 2D and 3D thermocapillary flows by a least-squares finite element method

“…The velocity vectors, on the cross-section of z = 0.5 for the thermocapillary flow, as shown in Figure 7, are similar to those for low Reynolds number lid-driven cavity flow. However, as shown in Figure 8, the velocity vector on the cross-section of y =0.5 are totally different from those for lid-driven cavity flows [14]. The temperature contours become more tortuous with time, as shown in It should be mentioned that the vortices at the corners of the free surface for Re= 100 and Pr =1.0 are not observed in the work by Babu and Korpela [8].…”

“…The main advantage of the LSFEM is that it leads to symmetric, positive definite (SPD) linear systems that can be solved efficiently by a matrix-free preconditioned conjugate gradient method, with minimum computer memory requirement for three-dimensional flows and transport processes. Recently, it has been proven that the LSFEM is a promising method for fluid flows and transport processes with Dirichlet boundary conditions [9][10][11][12][13][14][15]. Theoretical results using the LSFEM for a variety of differential equations have also been obtained [16][17][18][19].…”

The least-squares meshfree method for the steady incompressible viscous flow

“…It is simple to get the Jacobi preconditioned matrix. It needs not to form global stiffness matrix and the stiffness matrix at each evaluation point, so that computation cost can be reduced greatly [28]. It should be mentioned that the element-by-element technique is used in FEM because assemblage process is performed on the element level.…”

Least‐squares meshfree method for incompressible Navier–Stokes problems

“…It is simple to get the Jacobi preconditioned matrix. It needs not to form the global sti ness matrix at the evaluation points, so that computation cost can be reduced greatly [31]. It should be mentioned that the element-by-element technique is used in FEM because assemblage process is performed on the element level.…”

“…The matrix-free element-by-element Jacobi preconditioned conjugate gradient (MFEBEJCG) method is used to solve the resulting linear algebraic equations. The stopping criterion of MFEBEJCG iteration is given as Equation (31). The following criterion is used to judge when the steady-state solution is obtained:…”