Moresi, L.Quenette, S.Lemiale, V.Mériaux, C.Appelbe, B.Mühlhaus, H. B.2025-05-302025-05-300031-9201ORCID:/0000-0003-3685-174X/work/162950215http://www.scopus.com/inward/record.url?scp=34547874986&partnerID=8YFLogxKhttps://hdl.handle.net/1885/733754768We outline a mathematical formulation for mantle convection which can deal with the viscoelastic-plastic rheology of the cool parts of the lithosphere. This formulation is then analyzed to expose the numerical challenges inherent in the equations and a suitable solution strategy is outlined. With this strategy in place, we discuss a parallel implementation, explaining how we maintain computational efficiency, in a tiered and modular software environment. We show two examples from geodynamics which demonstrate the capability of the numerical method and its implementation.Algorithm development for lithospheric deformation models and plume conduit instabilities was supported by Australian Research Council Discovery projects DP0345157 and DP0449979. Funding for StGermain–Underworld and gLucifer has been predominantly provided by VPAC, Monash University, the ACceSS MNRF Snark and data assimilation projects, and the APAC CTT program. We would also like to thank the contributions from the Computational Infrastructure in Geodynamics, Caltech’s GeoFramework NSF ITR, and Deakin University’s Xanthus project. The VPAC development team also includes Patrick Sunter, Alan Lo, Luke Hodkinson, Raquibul Hassan, Kathleen Humble and John Spencer. The Monash development team also includes Julian Giordani, Dave May, Mirko Velic, Rob Turnbull and Cécile Duboz. Dave Stegman, Justin Freeman, Rebecca Farrington, Wendy Mason and Wendy Sharples have been influential contributors and valiant testers of this infrastructure.14enFinite elementMantle convectionMultigridObject-oriented methodsParallel computingParticle-in-cellPlasticityViscoelasticity2007-08-1510.1016/j.pepi.2007.06.00934547874986