Efficient Uzawa algorithms with projection strategies for geodynamic Stokes flow

Stokes equations are often difficult to handle in geodynamic modelling because they form a saddle-point system and involve strong variations in viscosity. Uzawa-type solvers are straightforward to implement, but their convergence may become slow if a suitable preconditioner is not used. Here, we introduce two adjustments that improve stability and efficiency. Residuals are evaluated in weak form, giving an effect similar to that of a mass-matrix preconditioner. We also add a projection step so that the velocity field remains nearly divergence-free. These updates made the solver converge faster and behave more stably than the standard Uzawa method. The modified approach was tested in several cases, including ABC flow, SolCx, mantle convection, block sinking, and compressible convection under the Anelastic Liquid Approximation.