Petrov-Galerkin

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

In the consistent streamline upwind Petrov-Galerkin (SUPG) scheme all of the terms in Equation (3.52) are weighted using the function defined by Equation (3.53) and hence Wjj = Wj. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Hughes, T. J. R., Franca, L. P. and Balestra, M., 1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition - a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolations. Cornput. Methods Appl. Meek Eng. 59, 85-99. 

Petera, J., Nassehi, V. and Pittman,. T.F.T., 1993. Petrov-Galerkin methods on... 

Least-square.s and streamline upwind Petrov-Galerkin (SUPG) schemes... 

Retaining all of the terms in the w eight function a least-squares scheme corresponding to a second-order Petrov-Galerkin formulation will be obtained. 

Petrov-Galerkin scheme - to discretize the energy Equation (5.25) for the calculation of T. 

Petera, J., Nassehi, V. and Pittman, J. F. T., 1993. Petrov Galerkin methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Int. J. Numer. Methods Pleat Fluid Flow 3, 205-222. 

Streamline upwind Petrov/Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,... 

In the finite element method, Petrov-Galerkin methods are used to minimize the unphysical oscillations. The Petrov-Galerkin method essentially adds a small amount of diffusion in the flow direction to smooth the unphysical oscillations. The amount of diffusion is usually proportional to Ax so that it becomes negligible as the mesh size is reduced. The value of the Petrov-Galerkin method lies in being able to obtain a smooth solution when the mesh size is large, so that the computation is feasible. This is not so crucial in one-dimensional problems, but it is essential in two- and three-dimensional problems and purely hyperbolic problems. 

Miller, C. T., and Rabideau, A. J. (1993). Development of split-operator Petrov-Galerkin methods for simulating transport and diffusion problems, Water Resources Research, 29(7), 2227-2240. 

In the mathematical literature, the Galerkin method is also known as Galerkin-Bubnov, while the case Wj / finite element formulations, such as those where the heat transfer is governed by convective effects. The application of Galerkin s method in the finite element method will be covered in detail in Chapter 9 of this textbook. 

There are many ways of solving the energy equation with convection effects. One that will be presented here is the widely accepted streamline upwind Petrov-Galerkin method... 

An analogous result is valid for continuous approximations of r when up winding is performed by the streamline upwinding Petrov-Galerkin method (SUPG) [104]. The same is true for finite element methods based on a quadrangular mesh [105]. 

An element for the stress components composed of 16 sub-elements (4x4) on which bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28]. This leads to a continuous C° approximation of the three variables. The velocity is approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([17]) made a mathematical analysis of the Stokes problem for this three-field formulation. They conclude that the polynomial approximations of the different variables should satisfy the generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they proved it was the case for the Marchal and Crochet element. In order to take into account the hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG). Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in [13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this problem, wiggles in the stress and the velocity field were obtained. In the SU method, the modified weighting function only applies to the convective terms in the constitutive equations. 

Also choose the Artificial Diffusion option and choose the Petrov-Galerkin method. This adds additional diffusion to the problem. Choose the Init tab and set the initial concentration to zero. 

The solution is a good representation of the solution, but the front, where the concentration drops quickly, is not as steep as it should be (Finlayson, 1992). The front is smoothed somewhat owing to the added diffusion term and the use of the Petrov-Galerkin method. If you solve the problem without either of these artifacts [e.g., D = 0 in Eq. (9.45) and using no Artificial Diffusion], the solution oscillates wildly, as seen in Figure 9.16. 

One way to fix this problem is to refine the mesh until Eq. (F.44) is satisfied, but for large problems that approach may require too many points (especially in 2D and 3D). Another way is to add some fake diffusion to the problem. In this case, the Petrov-Galerkin method is... 

This means that as Pe increases, the mesh size must decrease. Since the mesh size decreases, it takes more elements or grid points to solve the problem, and the problem may become too big. One way to avoid this is to introduce some numerical diffusion, which essentially lowers the Peclet number. If this extra diffusion is introduced in the flow direction only, the solution may still be acceptable. Various techniques include upstream weighting (finite difference [10]) and Petrov-Galerkin (finite element [11]). Basically, if a numerical solution shows imphysical oscillations, either the mesh must be refined, or some extra diffusion must be added. Since it is the relative convection and diffusion that matter, the Peclet number should always be calculated even if the problem is solved in dimensional units. The value of Pe will alert the chemist, chemical engineer, or bioengineer whether this difficulty would arise or not. Typically, is an average velocity, x is a diameter or height, and the exact choice must be identified for each case. 

If this problem is solved with a large Peclet number, oscillations appear (Ref [15] p. 217) and the mesh must be refined or some stabilization (like Petrov-Galerkin or upstream weighting) must be applied. The stabilization, of course, smoothes the solution and adds unphysical diffusion. It is up to the analyst to decide if that effect can be tolerated. 

Hughes, T. J. R. Brooks, A. N. (1982) Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations. Computer Methods in Applied Mechanics and Engineering il, 199-259. 

---