Then explicit methods could automatically use a solve to apply the mass 
matrix inverse for each "right hand side" application and implicit methods 
could automatically "drop in" the mass matrix with the shift when a right hand 
side Jacobian is given also. Instead of requiring the user to handle this 
themselves. Are there any gotcha's that I am missing or is this relatively 
straightforward?

   Why wouldn't we have this interface? 

   Note that adjoints are slightly tricky with a mass matrix without this API.

    Almost for sure MFEM must have this type of API.

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