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Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is also a first spatial derivative term. I saw an example of solving the convection diffusion equation by a FEM where only the second spatial derivative term was integrated by parts. If this is a normal practice, why is integration by parts not applied on the first spatial derivative term when solving a convection diffusion equation?

How about a set of PDEs, for example, Navier Stokes, where $\nabla{p}$ is integrated by parts?

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The purpose of doing integration by parts during trasforming to weak formulation is to lower 2nd order dirivatives to 1st order. If a derivative is already of 1st order, there's no need to do anything. 1st order derivatives are no problem for weak formulation.

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    $\begingroup$ Another thing are the boundary conditions. Both approaches are valid but you chose to lump up the convection term with the integration by parts depending on the Neumann boundary condition. $\endgroup$ Dec 24, 2022 at 18:05
  • $\begingroup$ thanks, is there such flexibility out of personal taste as to choose to lump up the convection term with the integration by parts or to choose not to do so based on the same Neumann BC? $\endgroup$
    – feynman
    Dec 25, 2022 at 3:08
  • $\begingroup$ is it allowed at all to also integrate first spatial derivatives by parts? What difference would that make? $\endgroup$
    – feynman
    Dec 25, 2022 at 3:13
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    $\begingroup$ You have to be guided by the physics of the problem in deciding whether to integrate a particular term by parts. If it results in a natural boundary condition that is not physically meaningful, then you should not do it. $\endgroup$ Dec 25, 2022 at 19:36
  • $\begingroup$ many thanks. Is variational/virtual work principle a good framework for deciding which bulk terms should be cast into surface terms by integration by parts, where surface terms are usually boundary displacement/force conditions? $\endgroup$
    – feynman
    Dec 26, 2022 at 4:01

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