Comparison of second-order accurate TVD scheme and p-version space-time least-squares finite-element method for nonlinear hyperbolic problems

2002 ◽  
Author(s):  
Ming Hseng Tseng ◽  
Shin-Jye Liang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Space Time ◽  
Second Order ◽  
Tvd Scheme ◽  
Hyperbolic Problems ◽  
Version Space ◽  
Element Method

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

2010 ◽  
Author(s):  
Rajeev Kumar ◽  
Brian H. Dennis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Least Squares Method ◽  
Second Order ◽  
Basis Functions ◽  
Governing Equations ◽  
First Order ◽  
Convection Dominated ◽  
Element Method

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.

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