Pragmatic solvers for 3D Stokes and elasticity problems with heterogeneous coefficients: evaluating modern incomplete LDLT preconditioners
Authors: Sanan Patrick , May Dave A. , Bollhöfer Matthias , Schenk Olaf 
Published: November 2020

Abstract. The need to solve large saddle point systems within computational Earth sciences is ubiquitous.
Physical processes giving rise to these systems include porous flow (the Darcy equations), poroelasticity, elastostatics, and highly viscous flows (the Stokes equations).
The numerical solution of saddle point systems is non-trivial since the operators are indefinite. Primary tools to solve such systems are direct solution methods (exact triangular factorization) or approximate block factorization (ABF) preconditioners.
While ABF solvers have emerged as the state-of-the-art scalable option, they are invasive solvers requiring splitting of pressure and velocity degrees of freedom, a multigrid hierarchy with tuned transfer operators and smoothers, machinery to construct complex Schur complement preconditioners, and the expertise to select appropriate parameters for a given coefficient regime – they are far from being “black box” solvers. Modern direct solvers, which robustly produce solutions to almost any system, do so at the cost of rapidly growing time and memory requirements for large problems, especially in 3D.
Incomplete LDLT (ILDL) factorizations, with symmetric maximum weighted-matching preprocessing, used as preconditioners for Krylov (iterative) methods,
have emerged as an efficient means to solve indefinite systems.
These methods have been developed within the numerical linear algebra community but have yet to become widely used in applications, despite their practical potential;
they can be used whenever a direct solver can, only requiring an assembled operator, yet can offer comparable or superior performance,
with the added benefit of having a much lower memory footprint.
In comparison to ABF solvers, they only require the specification of a drop tolerance and thus provide an easy-to-use addition to the solver toolkit for practitioners. Here, we present solver experiments employing incomplete LDLT factorization with symmetric maximum weighted-matching preprocessing to precondition operators
and compare these to direct solvers and ABF-preconditioned iterative solves.
To ensure the comparison study is meaningful for Earth scientists,
we utilize matrices arising from two prototypical problems, namely Stokes flow and quasi-static (linear) elasticity,
discretized using standard mixed finite-element spaces.
Our test suite targets problems with large coefficient discontinuities across non-grid-aligned interfaces,
which represent a common challenging-for-solvers scenario in Earth science applications.
Our results show that
(i) as the coefficient structure is made increasingly challenging, by introducing high contrast and complex topology with a multiple-inclusion benchmark, the ABF solver can break down, becoming less efficient than the ILDL solver before breaking down entirely;
(ii) ILDL is robust, with a time to solution that is largely independent of the coefficient topology and mildly dependent on the coefficient contrast;
(iii) the time to solution obtained using ILDL is typically faster than that obtained from a direct solve, beyond 105 unknowns; and
(iv) ILDL always uses less memory than a direct solve.