  Flow through porous, elastically deforming media is present in a variety of
natural contexts ranging from large-scale geophysics to cellular biology. In
the case of incompressible constituents, the porefluid pressure acts as a
Lagrange multiplier to satisfy the resulting constraint on fluid divergence.
The resulting system of equations is a possibly non-linear saddle-point problem
and difficult to solve numerically, requiring nonlinear implicit solvers or
flux-splitting methods. Here, we present a method for the simulation of flow
through porous media and its coupled elastic deformation. The pore pressure
field is calculated at each time step by correcting trial velocities in a
manner similar to Chorin projection methods. We demonstrate the method's
second-order convergence in space and time and show its application to phase
separating neo-Hookean gels.
