The challenge stems from the peculiar physics of the mantle. With a viscosity somewhere between 10^19 and 10^23 pascal-seconds — roughly a trillion trillion times thicker than water — mantle rock flows at what physicists call an extremely low Reynolds number, meaning inertia is utterly negligible. The Navier-Stokes equations that govern most fluid dynamics collapse into the simpler, elliptic Stokes system. But simpler physics does not mean easier computation. When researchers discretize these equations on a mesh using finite element methods, they obtain a so-called saddle-point system that couples velocity and pressure. That structure, combined with viscosity that can vary by many orders of magnitude across a single model domain, makes direct solvers impractically memory-hungry in three dimensions and pushes scientists toward iterative methods that converge only gradually under such harsh conditions.

A research team led by Deok-Kyu Jang and Byung-Dal So of Kangwon National University, together with colleagues at Utrecht University, revisited one of the oldest tools in the iterative solver toolbox: the Uzawa algorithm, first introduced in 1958. The classical approach eliminates the velocity variables and solves a reduced system for pressure alone, recomputing the velocity at each step. It is conceptually simple, but its convergence is sluggish and hypersensitive to a tuning parameter called the relaxation factor. In geodynamic applications, where rock viscosity can jump by factors of a million or more between adjacent regions, the standard method frequently stalls. The team’s key insight was to reformulate the entire iteration through the lens of finite element variational principles, changing how the algorithm measures its own errors.

Instead of computing residuals as raw algebraic vectors, the researchers redefined the error measurement using weighted inner products in the pressure space — a move that is mathematically equivalent to applying a pressure mass-matrix preconditioner. The result, they show, is that the Uzawa iteration becomes a preconditioned conjugate gradient method applied to the reduced Schur complement system. The team then pushed the idea further with a viscosity-weighted variant, in which the residual is measured under an inner product scaled by the inverse of the local viscosity. Because this weighting better mirrors the spectral properties of the true Schur complement in variable-viscosity problems, the resulting solver, called CD-U-η, converges faster and more stably precisely where traditional methods struggle most: across sharp viscosity contrasts that mimic subducting slabs and cratonic roots.

The second innovation addresses a subtler but equally consequential problem: mass conservation. Uzawa-type iterations suppress violations of the incompressibility constraint only asymptotically, so after a finite number of steps, the computed velocity field still drifts slightly from being divergence-free. In a single time step, the error is tiny — but in mantle convection simulations that run for thousands of steps, these small errors accumulate and contaminate the transported fields that scientists actually care about, such as temperature, composition, and density. The team’s remedy is a post-processing projection step based on the discrete Helmholtz-Hodge decomposition, a classical technique from unsteady fluid dynamics that had rarely been applied to elliptic Stokes problems in geodynamics. By solving an additional scalar Poisson problem, the method adds a corrective gradient field that nudges the velocity back onto the constraint-satisfying subspace.

The beauty of the projection, the authors emphasize, lies in its cost profile. Each additional Uzawa iteration demands a fresh solve of the expensive momentum equation, whereas the projection requires only a single, comparatively cheap Poisson solve per time step. In their benchmarks, the combined overhead of the new residual formulation and the projection corrections amounted to less than 5 percent of total solver time, because the momentum solves — which the viscosity-weighted formulation actually reduces in number — remain the dominant expense. In one demanding block-sinking test, the viscosity-weighted solver cut the average number of momentum solves per time step from 22.2 to 8.35, slashing total solver runtime from 4168 seconds to 1735 seconds on 32 processor cores.

The benchmarks themselves read like a tour of computational geodynamics’ standard gauntlet. The team tested their solvers on the ABC flow, an idealized manufactured solution with smooth analytical answers; on SolCx and SolKz, canonical benchmarks featuring viscosity contrasts of up to a million; on the multisinker problem, which embeds eight high-viscosity blobs in a soft medium with dynamic viscosity ratios reaching an extraordinary 10^10; and on full mantle convection scenarios, both incompressible and compressible under the Anelastic Liquid Approximation. Across the suite, the conjugate-directions solver with viscosity weighting converged rapidly and monotonically where the standard and adaptive variants stagnated, and a mesh refinement study confirmed that the method retains the optimal third-order accuracy in velocity and second-order accuracy in pressure expected of Taylor-Hood finite elements.

Importantly, the implementation was built entirely from standard components: the open-source FEniCS finite element library coupled with the PETSc parallel solver toolkit, using conjugate gradient and GMRES iterations preconditioned by algebraic multigrid. No custom low-level code was required, which means the approach is portable, reproducible, and readily transferable to other platforms — a significant practical consideration for a community whose flagship simulation codes, such as CitcomS and ASPECT, already rely on Uzawa-type updates and Schur complement preconditioning. Because the entire algorithm is expressed through inner products and variational forms, it slots naturally into the matrix-assembly and parallel-solver infrastructure that general-purpose finite element libraries already provide.

The long-term payoff of the projection step was particularly striking in time-dependent experiments. Over 5000 time steps of linear mantle convection with temperature- and depth-dependent viscosity, the projection kept velocity errors smaller at every stage and, because temperature and composition fields are carried along by the velocity, those fields accumulated the benefits as well. Localized errors near material interfaces — a persistent headache in multiphase problems like the block-sinking benchmark, where a dense block dives through a denser-contrasting medium — were visibly reduced, bringing the iterative solution into closer agreement with a direct solver treated as ground truth. In a three-dimensional thermochemical convection test, the projection roughly halved the divergence error throughout the domain, including near rising plume structures.

The authors are candid about the limits of their approach: the projection corrects only the velocity field and does not directly improve pressure convergence, which remains governed by the underlying Uzawa iteration — a limitation that matters because pressure plays a critical role in many geodynamic applications. Still, the study offers the geodynamics community a complementary algorithmic perspective on a problem it has wrestled with for decades. As simulations grow toward exascale resolutions and incorporate ever more realistic rheologies, even modest per-step savings compound into transformative gains. With both refinements available at nearly negligible extra cost, projection-enhanced, viscosity-weighted Uzawa solvers may well become a new baseline for the computational engines that let us peer, in silico, into the ro machinery of Earth’s deep interior.

Subject of Research: Efficient Uzawa algorithms with projection strategies for solving geodynamic Stokes flow with variable viscosity

Article Title: Efficient Uzawa algorithms with projection strategies for geodynamic Stokes flow

Article References: Jang, D.-K., Lee, K.-M., Thieulot, C., Choi, W.-H., & So, B.-D. (2026). Efficient Uzawa algorithms with projection strategies for geodynamic Stokes flow. Solid Earth, 17(10), 1167-1185. https://doi.org/10.5194/se-17-1167-2026

Image Credits: AI Generated

DOI: 10.5194/se-17-1167-2026

Keywords: geodynamics, Stokes flow, Uzawa algorithm, mantle convection, finite element method, Schur complement, preconditioned conjugate gradient, Helmholtz-Hodge projection, viscosity contrast, numerical modeling, PETSc, FEniCS