Petrov-Galerkin method

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. 

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). 

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

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. 

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. 

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]. 

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... 

Method of weighted residuals - Petrov—Galerkin Method... 

A well known solution technique is the Method of Weighted Residuals also referred as Petrov-Galerkin method. Let Ym and Zm be finite dimensional subspaces of H spanned by yi,...,Pm and respectively. 

In terms of a Petrov-Galerkin method the single trial (and test) space is replaced by a sequence of nested subspaces V = Vj j>o such that Uj>o is dense in Ji. These spaces are often given in terms of their bases, i.e. ... 

Petrov-Galerkin method with basis functions 1 — J and /2K Vo/... 

Crochet, M.J. and Legat, V. (1992) The consistent streamline upwind Petrov-Galerkin method for viscoelastic flow revisited. J. Non-Neivtonian Fluid Mech., 42, 283-299. 

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