Petrov-Galerkin technique

FeUceUi et al. [57] extended the model to three-dimensional problems and applied it to solidification of a Pb-Sn alloy in a parallelepiped and in a cylinder. The transport equations were discretized and integrated in time using a FEM based on the bilinear Lagrangian isoparametric element. A Petrov-Galerkin technique was used to... 

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). 

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. 

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. 

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