  The objective of this paper is twofold. First, we propose two composable
block solver methodologies to solve the discrete systems that arise from finite
element discretizations of the double porosity/permeability (DPP) model. The
DPP model, which is a four-field mathematical model, describes the flow of a
single-phase incompressible fluid in a porous medium with two distinct
pore-networks and with a possibility of mass transfer between them. Using the
composable solvers feature available in PETSc and the finite element libraries
available under the Firedrake Project, we illustrate two different ways by
which one can effectively precondition these large systems of equations.
Second, we employ the recently developed performance model called the
Time-Accuracy-Size (TAS) spectrum to demonstrate that the proposed composable
block solvers are scalable in both the parallel and algorithmic sense.
Moreover, we utilize this spectrum analysis to compare the performance of three
different finite element discretizations (classical mixed formulation with
H(div) elements, stabilized continuous Galerkin mixed formulation, and
stabilized discontinuous Galerkin mixed formulation) for the DPP model. Our
performance spectrum analysis demonstrates that the composable block solvers
are fine choices for any of these three finite element discretizations. Sample
computer codes are provided to illustrate how one can easily implement the
proposed block solver methodologies through PETSc command line options.
