[dune-pdelab] Why the lambda_*** methods?

[Oliver Sander]

Dear pdelab team,

I have recently tried to explain to somebody why the lambda_*** methods
are separate in the LocalOperator interface, rather than having alpha_***
assemble the entire residual.  As it turns out, I am not sure I know
the reason myself.

Initially, I thought this was to make Newton solvers more efficient.
A Newton solver needs to call alpha_*** at each iteration, but lambda_***
only once.  However, the Newton implementation in dune-pdelab calls
lambda_*** at each iteration.  In fact, there is no way to access the
alpha_*** contribution throught the GridOperator interface at all!
So that's not the reason.

Then I learned that the finite difference code in NumericalJacobian uses
only alpha_***, but disregards lambda_***.  That makes sense: in first-order
FD formulas, the lambda_*** parts cancel out anyway.

But now I am wondering: Is that really all?  Because that reason does
not really convince me:

- While having lambda_*** separate may speed up FD approximations of the
  Jacobian, it seems likely that it will slow down other things.
  For example, calling lambda_*** and alpha_*** together means that
  shape functions have to be computed twice, there are two separate
  quadrature loops, etc.

- Having separate lambda_*** methods makes LocalOperator implementations
  much larger (statically).  Adding a normal source term to a Poisson assembler
  is simply two extra lines in alpha_volume.  A separate lambda_volume
  method is at least 20 lines.

- Its doubtful whether FD approximations are desirable to begin with:
  Algorithmic differentiation is faster and more precise.

Can I please get some insight from the pdelab team?  Am I missing something?

Thanks,
Oliver

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