  We present a new hybrid semi-implicit finite volume / finite element
numerical scheme for the solution of incompressible and weakly compressible
media. From the continuum mechanics model proposed by Godunov, Peshkov and
Romenski (GPR), we derive the incompressible GPR formulation as well as a
weakly compressible GPR system. As for the original GPR model, the new
formulations are able to describe different media, from elastoplastic solids to
viscous fluids, depending on the values set for the model's relaxation
parameters. Then, we propose a new numerical method for the solution of both
models based on the splitting of the original systems into three subsystems:
one containing the convective part and non-conservative products, a second
subsystem for the source terms of the distortion tensor and heat flux equations
and, finally, a pressure subsystem. In the first stage of the algorithm, the
transport subsystem is solved by employing an explicit finite volume method,
while the source terms are solved implicitly. Next, the pressure subsystem is
implicitly discretised using finite elements. Within this methodology,
unstructured grids are employed, with the pressure defined in the primal grid
and the rest of the variables computed in the dual grid. To evaluate the
performance of the proposed scheme, a numerical convergence analysis is carried
out, which confirms the second order of accuracy in space. A wide range of
benchmarks is reproduced for the incompressible and weakly compressible cases,
considering both solid and fluid media. These results demonstrate the good
behaviour and robustness of the proposed scheme in a variety of scenarios and
conditions.
