Mathematics

The aim of this work is to solve parametrized partial differential equations in computational
domains represented by networks of repetitive geometries by combining reduced
basis and domain decomposition techniques. The main idea behind this approach is to
compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary
conditions on the boundaries: these functions will represent the basis of a reduced space
where the global solution is sought for. The continuity of the latter is assured by a classical
domain decomposition approach. Test results on Poisson problem show the
flexibility of
the proposed method in which accuracy and computational time may be tuned by varying
the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems.
Thanks to this feature, it allows dealing with arbitrarily complex network and features
more flexibility than a classical global reduced basis approximation where the topology of
the geometry is fixed.