Efficient Uzawa algorithms with projection strategies for geodynamic Stokes flow
- 1Kangwon National University (Chuncheon, Korea, Republic of)
Abstract
Stokes equations are often difficult to handle in geodynamic modelling because they form a saddle-point system and involve strong variations in viscosity. In this work, we reformulate the Uzawa iteration through finite element variational principles, replacing the standard algebraic residual with a weighted L ² form that yields a robust preconditioned conjugate gradient iteration for the Schur complement system. 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. This reformulation provides a complementary algorithmic perspective on Schur complement preconditioning within the finite element variational framework. The modified approach was tested in several cases, including ABC flow, SolCx, SolKz, mantle convection, block sinking, multisinker, and compressible convection under the Anelastic Liquid Approximation.
Versions
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BibTeX
@article{jang2026efficient,
title = {{Efficient Uzawa algorithms with projection strategies for geodynamic Stokes flow}},
author = {Deok-Kyu Jang and Kyeong-Min Lee and Cedric Thieulot and Whan-Hyuk Choi and Byung-Dal So},
journal = {Solid Earth},
year = {2026},
month = oct,
doi = {10.5194/se-17-1167-2026},
url = {https://se.copernicus.org/articles/17/1167/2026/},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Efficient Uzawa algorithms with projection strategies for geodynamic Stokes flow
AU - Deok-Kyu Jang
AU - Kyeong-Min Lee
AU - Cedric Thieulot
AU - Whan-Hyuk Choi
AU - Byung-Dal So
PY - 2026
DA - 2026/10/19
JO - Solid Earth
DO - 10.5194/se-17-1167-2026
UR - https://se.copernicus.org/articles/17/1167/2026/
AB - Stokes equations are often difficult to handle in geodynamic modelling because they form a saddle-point system and involve strong variations in viscosity. In this work, we reformulate the Uzawa iteration through finite element variational principles, replacing the standard algebraic residual with a weighted L ² form that yields a robust preconditioned conjugate gradient iteration for the Schur complement system. 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. This reformulation provides a complementary algorithmic perspective on Schur complement preconditioning within the finite element variational framework. The modified approach was tested in several cases, including ABC flow, SolCx, SolKz, mantle convection, block sinking, multisinker, and compressible convection under the Anelastic Liquid Approximation.
SN - 1869-9529
ER -