Constitutive laws for viscoplastic materials involve a multivalued nonlinearity. Duality methods for such equations are known to converge, but the convergence is slow, requiring a large number of iterations. We introduce here a new iterative method of implicit primal-dual type for a class of variational problems, with a particular asymmetrical form in terms of the primal and dual unknowns, with an exact resolution of one relation and an approximate resolution of the other (semi-exact method). Two fast algorithms are proposed. The first includes an adaptive choice of a variable parameter, and the second includes a Newton-like linearized correction procedure. Applying the semi-exact method to a steady viscoplastic problem leads at each iteration to the resolution of a Laplace type equation, thus equivalent in terms of cost per iteration to the well-known Augmented Lagrangian or dual FISTA methods. Numerical tests show that the semi-exact method achieves a faster convergence in terms of number of iterations, compared to the augmented Lagrangian method or to the FISTA* method with acceleration, for Bingham or Herschel–Bulkley laws. This is particularly true when a large zero order term is present, as is the case when solving a time dependent problem.
Keywords: Viscoplastic materials, semi-exact method, implicit primal-dual algorithm, adaptive parameter, Newton linearized correction, steady flows
François Bouchut 1 ; David Maltese 1
@article{SMAI-JCM_2026__12__135_0,
author = {Fran\c{c}ois Bouchut and David Maltese},
title = {A semi-exact primal-dual method for steady viscoplastic flows},
journal = {The SMAI Journal of computational mathematics},
pages = {135--170},
year = {2026},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {12},
doi = {10.5802/jcm.145},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/jcm.145/}
}
TY - JOUR AU - François Bouchut AU - David Maltese TI - A semi-exact primal-dual method for steady viscoplastic flows JO - The SMAI Journal of computational mathematics PY - 2026 SP - 135 EP - 170 VL - 12 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/jcm.145/ DO - 10.5802/jcm.145 LA - en ID - SMAI-JCM_2026__12__135_0 ER -
%0 Journal Article %A François Bouchut %A David Maltese %T A semi-exact primal-dual method for steady viscoplastic flows %J The SMAI Journal of computational mathematics %D 2026 %P 135-170 %V 12 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/jcm.145/ %R 10.5802/jcm.145 %G en %F SMAI-JCM_2026__12__135_0
François Bouchut; David Maltese. A semi-exact primal-dual method for steady viscoplastic flows. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 135-170. doi: 10.5802/jcm.145
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