Pasha, 

there are tutorial programs, such as step-22[1], that use BlockMatrices and 
solve for individual blocks.
What exactly is the problem you are facing? Does it have anything to do 
with Trilinos objects or is it a conceptual problem?

Can you tell us how exactly you want to split solving your system?

Best,
Daniel

[1] https://www.dealii.org/8.4.0/doxygen/deal.II/step_22.html

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