Classical Galerkin Finite Element method (GFEM) which is more suited for self-adjoint type system of equations as in solid mechanics and heat conduction struggles when applied to non-self-adjoint systems, as encountered in fluid dynamics. Velocity and pressure variables must be approximated using functions that belong to different spaces and must satisfy the stringent LBB condition in order to avoid a saddle point problem. The resulting system matrix is asymmetric and non-positive definite.
The least-squares finite element method (LSFEM), which is based on minimizing the l2-norm of the residual, is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. It has many attractive advantages over the Galerkin finite element method such as the lack of an inf-sup condition and the resulting symmetric positive definite system of algebraic equations, unlike GFEM, which make LSFEM an ideal approach for fluid dynamic equations. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have limited the application of LSFEM for large-scale practical problems, such as three-dimensional compressible viscous flows.
A simple yet novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second-order derivatives in the governing equations, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. We call this method Least-Squares/Galerkin Split Finite Element Method (LSGSFEM). When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. The resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers such as the preconditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with results of several benchmark problems solved using low-order C0 continuous elements, both in incompressible and compressible flow regimes.