The Least-Squares Galerkin Split Finite Element Method for Buoyancy-Driven Flow

The least-squares finite element method (LSFEM), 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. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike Galerkin finite element method (GFEM). 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 computational requirements that have limited the application of LSFEM to large-scale practical problems. A novel finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The method has been successfully applied here to solve complex buoyancy-driven flow with Boussinesq approximation in a square cavity with differentially heated vertical walls using low-order C⁰ continuous elements.