Composable block solvers for the four-field double porosity/permeability model

TL;DR

This paper introduces two composable block solver methods for the double porosity/permeability model, demonstrating their scalability and effectiveness across different finite element discretizations using PETSc and Firedrake.

Contribution

It presents novel composable block solver strategies for the DPP model and evaluates their scalability and performance across multiple discretizations using the TAS spectrum.

Findings

Solvers are scalable in parallel and algorithmic sense.

Performance spectrum analysis guides discretization choice.

Sample codes facilitate implementation of proposed methods.

Abstract

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.