Judy:

(1) Is there an internal command/function that outputs higher-order-elemental (Q2, Q3 elements) nodal physical coordinate?
Just like, "cell->vertex" outputs grid points that works for Q1 element.
If I want to count in a linear function f(x) = x on Laplace's right-hand-side, should I just manually make an expression, to interpolate mid-nodes, as well as other fractional-nodes @ 1/3 and 2/3 of each cell "he", to construct “x”?

You can query the location of all degrees of freedom via the DoFTools::map_dofs_to_support_points() function. But I don't understand the connection to your right hand side: you generally only ever need to evaluate the right hand side function at *quadrature points*, not at *node points*. You get the physical locations of quadrature points via
  fe_values.quadrature_point(q_point)
as shown in several of the tutorial programs.


(2) In the nonlinear problem, are nodal physical coordinates updated thru finite element program, like, including "phi_i * x_i" or “grad_phi_i * x_i” for any i of each cell?

The library doesn't do anything unless you instruct it to, because it has no idea what kind of equation you are solving. So, unless you explicitly say "update the node location with this kind of information", nothing will happen.

Best
 W.

--
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Wolfgang Bangerth          email:                 bange...@colostate.edu
                           www: http://www.math.colostate.edu/~bangerth/

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