We consider the incompressible Stokes equations relevant to mantle convection. Because the Prandtl number of mantle materials is effectively infinite (Pr ≈ 10²³), inertial terms in the Navier–Stokes equations can be neglected, and the governing system reduces to the elliptic Stokes equations .

In a bounded domain Ω ⊂ Rnd (nd = 2, 3), these are ∇⋅σ+ρg=0,∇⋅u=0, where u is velocity, p is pressure, ρ is density, and g is gravitational acceleration. The Cauchy stress tensor is σ=-pI+2ηD(u),D(u)=12∇u+∇uT, with η the dynamic viscosity and D(u) the strain-rate tensor.

In compact form, the Stokes equations then become ∇⋅η∇u+∇uT-∇p+ρg=0,∇⋅u=0. In the case of constant viscosity with Dirichlet boundary conditions, the momentum equation reduces to a vector Laplacian.

Discretization with stable finite element pairs gives the saddle-point system KGGT0up=fh, where K is the discrete viscosity matrix (symmetric positive definite) and G is the discrete gradient operator. The system is symmetric but indefinite, and efficient iterative solvers are required.

Eliminating velocity gives the Schur complement system Sp=GTK-1f-h,S=GTK-1G. The Schur complement S is symmetric positive definite under the standard inf-sup stability condition. Forming S explicitly is too expensive, since each multiplication requires solving with K-1, which is costly for large problems with variable viscosity. In practice, S is applied implicitly within iterative methods, together with suitable preconditioners.

The standard Uzawa algorithm applies Richardson-type iterations to update the pressure and recompute the velocity, with relaxation parameter ω > 0. The scheme is simple, but its convergence is sensitive to the choice of ω and deteriorates with large viscosity contrasts.

To reduce this sensitivity, the relaxation parameter is updated adaptively at each step. This improves robustness compared with the fixed-ω method.

The CD-U method accelerates convergence by constructing conjugate directions, in analogy with the conjugate gradient method. The explicit coefficients αk and βk highlight the connection with CG. By exploiting information from previous steps, CD-U achieves faster and more stable convergence, especially for problems with strong viscosity contrasts or heterogeneous coefficients.

The definition of the residual is central to Uzawa-type algorithms. In the Schur complement formulation, the discrete residual is qk=h-GTuk, which measures the violation of the incompressibility constraint. In its standard (vector) form, qk is assembled directly as a discrete vector. This is simple but suffers from poor conditioning, and small singular values of GT can lead to unstable step-size choices and slow convergence, especially with large viscosity contrasts.

An alternative is to define the residual variationally in the L2 inner product on the pressure space Q: 1〈qk,r〉=〈h+∇⋅uk,r〉,∀r∈Q. In discrete form this corresponds to solving a mass-matrix system, Mpqk=Mph+b(uk), where b(uk) denotes the discrete constraint term. This formulation shows that Uzawa iterations implicitly apply the pressure mass matrix as a preconditioner . The resulting L2-projected residual is smoother and scale-independent, and yields more robust convergence at modest extra cost.

With this modification, the overall update structure of the standard, adaptive, and CD-U algorithms remains the same. Only the scalar coefficients change, as summarized below.

Standard Uzawa. 2〈pk+1,r〉=〈pk,r〉-ω〈qk+1,r〉,∀r∈Q.

Adaptive Uzawa. Using the L2 residual definition in Eq. (), the step size is updated as 3ωk+1=〈qk+1,qk+1〉〈∇qk+1,zk+1〉.

Conjugate-directions Uzawa (CD-U). With qk defined in Eq. (), the coefficients are 4αk=〈qk,qk〉〈∇dk,zk〉,βk=〈qk+1,qk+1〉〈qk,qk〉.

Conjugate-directions Uzawa with η-weighted residuals (CD-U-η). The standard L2 residual in Eq. () can be extended by replacing the mass matrix with an inverse-viscosity-weighted version, which we refer to as η-weighted for brevity. This introduces a viscosity scaling in the pressure space, which more closely reflects the spectrum of the true Schur complement for variable-viscosity problems. The resulting scheme is a preconditioned conjugate-gradient iteration in which both the residual qk and its preconditioned form wk appear in the update formulas.

This iteration can be interpreted as a Preconditioned Conjugate Gradient (PCG) method applied to the Schur complement system. Here qk denotes the residual in the pressure space, and wk its preconditioned form. The search directions dk and coefficients αk, βk are then defined from inner products such as 〈qk,wk〉, following the standard PCG structure.

With the L2 formulation, the preconditioner is the pressure mass matrix Mp, which is spectrally close to the identity. In this case wk = qk, and the scheme reduces to a Conjugate Gradient iteration with a trivial preconditioner. In the η-weighted formulation, the preconditioner is the viscosity-weighted mass matrix, so that wk ≠ qk and both quantities appear in the iteration. Note that the relation between qk and wk cannot be expressed pointwise as wk=ηqk, since η(x) varies spatially. Instead, wk is defined variationally by 〈1ηwk,r〉=〈qk,r〉,∀r∈Q, which ensures that wk belongs to the pressure space Q and serves as the Riesz representative of qk under the η-weighted inner product. This formulation provides a consistent SPD preconditioner in the finite element setting.

In practice, at high resolution or extreme viscosity contrasts, the conjugate gradient iteration may stagnate when the denominators in the step-size and conjugacy coefficients become too small. To address this, we introduce an adaptive scaling parameter AAR that modifies the preconditioner weights from 1/η(x) to AAR / η(x). Starting from AAR = 1.0, the parameter is progressively reduced whenever stagnation is detected. To prevent numerical instability in high-viscosity regions where the scaled weights become excessively small, we enforce a pointwise lower bound min⁡x(AAR/η(x))≥ϵ for a prescribed threshold ϵ. The specific reduction strategy and the threshold ϵ are reported together with the numerical experiments.

In summary, while the L2 formulation corresponds to a CG iteration with a trivial mass-matrix preconditioner, the η-weighted Uzawa method is a genuine PCG scheme with a physically motivated preconditioner that better reflects the spectral properties of the Schur complement, particularly under strong viscosity contrasts . The adaptive AAR strategy ensures robust convergence of this scheme across a wide range of resolutions and viscosity ratios.

Uzawa-type iterations reduce the constraint violation only asymptotically, so the discrete constraint remains slightly violated after a finite number of steps. To enforce the constraint explicitly, we apply a projection step as a post-processing. The idea is to correct an intermediate velocity u^ by adding a gradient field so that the updated velocity satisfies the discrete constraint operator C(u)=h, where C(u) = ∇⋅u for the incompressible case, and C(u) = ∇⋅(ρ‾u) under the anelastic liquid approximation (ALA).

For the incompressible case, C(u) = ∇⋅u, the projection requires solving a scalar Poisson problem -Δϕ=h+∇⋅u^,uk+1=u^+∇ϕ. This corresponds to the discrete Helmholtz–Hodge decomposition, which states that any vector field can be uniquely decomposed into a solenoidal part and a gradient part, u*=u+∇ϕ,∇⋅u=0. Equivalently, the projection operator can be written as P=I-G(GTG)-1GT, which is symmetric and idempotent.

In variational form, the correction is obtained by solving 〈∇ϕ,∇ψ〉=〈h+∇⋅u^,ψ〉,∀ψ∈Q, and updating the velocity consistently in the finite element space, 〈uk+1,w〉=〈u^,w〉+〈∇ϕ,w〉,∀w∈V. This weak formulation is important: a direct discrete update uk+1 = u^ + Gϕ does not yield the same accuracy, whereas the variational form ensures consistency and better convergence. In practice, the projection can also be implemented component-wise to avoid assembling a large coupled system.

For the compressible case, C(u) = ∇⋅(ρ‾u), and the projection takes the weighted form -∇⋅ρ‾∇ϕ=h+∇⋅ρ‾u^,uk+1=u^+∇ϕ, so that ∇⋅(ρ‾uk+1) = h is satisfied exactly. This weighted projection is the natural extension of the Helmholtz–Hodge decomposition to density-dependent flows.

Although projection methods of this type are standard in computational fluid dynamics, they have not been used previously as post-processing corrections in geodynamics. As we show in Sect. 3, this step reduces accumulated divergence errors and improves the long-term accuracy of mantle convection benchmarks.

A related approach in particle-in-cell methods is the conservative velocity interpolation (CVI) scheme , which enforces local mass conservation during interpolation but follows a different formulation from the projection-based correction used here. The CVI approach is specifically designed for quadrilateral Q1P0 elements, whereas our projection method is element-independent and applicable to general finite element discretizations.

We implemented the proposed algorithms using the open-source finite element library FEniCS (version 2019.2.0.13) . FEniCS provides a Python interface that simplifies variational form expression and automates finite element matrix assembly, which is useful in geodynamics where PDE systems often involve strongly variable rheologies . To solve the large sparse linear systems from discretizing the Stokes equations, we coupled FEniCS with PETSc . We used GMRES and CG with algebraic multigrid (AMG) preconditioning. Solver configurations were adjusted through PETSc runtime parameters.

All algorithms were implemented within the standard FEniCS–PETSc environment without custom code. This portability shows that the approach is reproducible, extendable, and transferable to other platforms, in line with current practices in large-scale geodynamic modeling.

Each Uzawa iteration with projection requires solving three linear systems: (i) the momentum equation for velocity, (ii) an L2-projection in the pressure space for residual representation and scalar potential projection, and (iii) a Poisson system for velocity correction.

The L2-projection system was solved with CG preconditioned by block Jacobi, which works well for the diagonally dominant mass matrices from finite element discretizations. The Poisson system for velocity correction was also solved with CG and BoomerAMG. All solvers used relative and absolute tolerances of 10-10 and 10-12, so that the convergence behavior observed in our benchmarks is governed by the outer Uzawa iteration rather than by inner solver accuracy. For reference, the ASPECT code uses default inner solve tolerances of 10⁻² for the velocity block and 10⁻⁶ for the Schur complement block.

Uzawa iterations were terminated using a normalized divergence residual Rdiv=‖∇⋅uk‖L2‖uk‖L2, which measures how well the continuity constraint is satisfied. We used relaxed thresholds in long mantle convection runs to balance efficiency and accuracy.

All benchmarks were discretized with Taylor–Hood (P2-P1) finite elements. Geodynamic models often use quadrilateral or hexahedral meshes, but we used structured triangular and tetrahedral meshes generated by splitting each cell along its diagonals. In two dimensions, this diagonal subdivision yields a symmetric “union-jack” pattern that preserves benchmark symmetry. This configuration provides robust velocity-pressure stability and straightforward implementation in FEniCS. Further discussion on finite element selection for geodynamic Stokes modeling can be found in . Viscosity η is represented as a finite-element field and evaluated at quadrature points during local assembly of element matrices.

To evaluate accuracy, we report the relative L2 errors: Eu=‖u-uexact‖L2‖uexact‖L2,Ep=‖p-pexact‖L2‖pexact‖L2.

We first establish a baseline solver among the three Uzawa variants: standard Uzawa (S-U), adaptive Uzawa (A-U), and conjugate-directions Uzawa (CD-U). For this comparison, we use two complementary tests. The ABC flow with constant viscosity assesses convergence behavior in an ideal setting and is used to compare S-U, A-U, and CD-U directly. The SolCx problem with a viscosity contrast of 10⁶ provides a stress test under strong heterogeneity; here we focus on CD-U and contrast three residual formulations – vector (algebraic), L2 inner-product, and the viscosity-weighted CD-U-η – to examine robustness. In both tests, we track the relative L2 errors in velocity and pressure and the normalized divergence residual over iterations, and select the most effective configuration for the remainder of the paper.

We next examine the effect of the post-processing correction in two incompressible tests: linear mantle convection with variable viscosity and the block sinking problem with sharp density and viscosity contrasts. In both tests the mesh, boundary conditions, and time stepping are kept identical; the only change is in the Stokes solver. Results from CD-U-η are compared with those from CD-U-η combined with post-processing. Reference solutions are obtained with the direct LU solver MUMPS , which solves the discrete system to near machine precision.

Performance is evaluated by the relative L2 errors of velocity and scalar fields (temperature, composition, or density) and by the normalized continuity residual Rdiv. A breakdown of solver cost is also reported, showing that the post-processing step adds little to the total runtime, while momentum solves remain the dominant cost.

We next consider mantle convection under the Anelastic Liquid Approximation (ALA) , which incorporates depth-dependent density while filtering out fast acoustic modes. This benchmark follows with Rayleigh number Ra = 10⁵ and dissipation number Di = 0.5. This case extends the projection-based correction to compressible formulations, where mass conservation is weighted by the background density profile.

Under ALA, mass conservation is expressed as ∇⋅(ρ‾u)=0,ρ‾(z)=ρ0exp⁡Di(1-z)γ0. The momentum equation is ∇⋅τ(u)-∇p′+Raρ‾αTk^-(Diγ0cpcv)ρ‾χTp′k^=0, where τ(u)=2με(u)-23μ(∇⋅u)I,ε(u)=12(∇u+∇u⊤).

The temperature equation, including advection, diffusion, adiabatic heating, and viscous dissipation, is ρ‾cp∂tT+ρ‾cpu⋅∇T-∇⋅(κ∇(T‾+T))+αρ‾DiuyT-DiRaσ(u):∇u=0, with the background profile T‾(z)=T0exp⁡(Di(1-z)).

Time discretization used a second-order Crank–Nicolson scheme. Since viscosity is constant, CD-U and CD-U-η coincide; the focus here is therefore on testing the robustness of the projection-based correction under compressible mass conservation.

To enforce the weighted continuity constraint after each Uzawa iteration, a ρ‾-weighted velocity correction was applied. Given an intermediate velocity u^, we solved -∇⋅(ρ‾∇ϕ)=∇⋅(ρ‾u^),u=u^+∇ϕ, ensuring ∇⋅(ρ‾u)=0 at the discrete level.

The computational domain Ω = [0,1]2 was discretized into 160 × 160 triangular Taylor–Hood (P2-P1) elements for velocity and pressure, and P1 discontinuous Galerkin elements for temperature. Viscosity was constant (η = 1). Free-slip boundaries were imposed on the horizontal sides and no-slip on the vertical sides. The temperature was fixed at the bottom (T ≈ 0.09) and top (T = 0), with insulated sidewalls. Initial conditions included sinusoidal perturbations of the background temperature, and pressure was initialized to zero. The Uzawa method with conjugate directions (CD-U) was used as the baseline solver, with convergence tolerance Rdivρ‾ < 10⁻³, and was compared against CD-U with projection-based correction. Reference solutions were computed using the direct solver MUMPS.

Figure  shows that with constant viscosity the CD-U solver alone is already robust, but the ρ‾-weighted correction further reduces errors in velocity and temperature. Figure  confirms improved accuracy near steep thermal gradients.

In terms of cost, the constant-viscosity ALA setup required only about 2–3 momentum solves per step on average, making it considerably cheaper than variable-viscosity cases. The ρ‾-weighted correction added less than 20 % to the runtime, confirming that its accuracy benefits come at modest overhead.