An Arbitrary-Order Augmented Lagrangian Discrete De Rham Scheme for Contact Mechanics with Tresca Friction
Abstract
We design and analyse an arbitrary-order augmented Lagrangian Discrete de Rham polytopal scheme for contact mechanics in fractured media. The fractures are represented as objects of codimension one in the matrix, a linear elastic medium, and a Tresca friction model is adopted when contact between fracture walls occurs. Both the normal Signorini-type/traction constraint and the Tresca friction law are recast as projection identities, and are combined into a single augmented Lagrangian bilinear form. The displacement is discretised by a DDR space of arbitrary polynomial degree k≥ 0, enriched on the fracture faces by bubble degrees of freedom that ensure stability via the existence of a discrete lifting between the Lagrange multiplier space and the discrete jump of the displacement along the fracture. We establish an abstract error estimate yielding optimal order convergence for the displacement and the Lagrange multiplier variables. Extensive numerical experiments confirm the theoretical analysis and demonstrate the practical accuracy and robustness of the scheme.
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BibTeX
@misc{droniou2026arbitraryorder,
title = {{An Arbitrary-Order Augmented Lagrangian Discrete De Rham Scheme for Contact Mechanics with Tresca Friction}},
author = {Jérôme Droniou and Raman Kumar and Roland Masson},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32471},
eprint = {2610.32471},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05836504v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - An Arbitrary-Order Augmented Lagrangian Discrete De Rham Scheme for Contact Mechanics with Tresca Friction
AU - Jérôme Droniou
AU - Raman Kumar
AU - Roland Masson
PY - 2026
DA - 2026/10/23
PB - HAL
DO - 10.48550/arxiv.2610.32471
UR - https://hal.science/hal-05836504v1
AB - We design and analyse an arbitrary-order augmented Lagrangian Discrete de Rham polytopal scheme for contact mechanics in fractured media. The fractures are represented as objects of codimension one in the matrix, a linear elastic medium, and a Tresca friction model is adopted when contact between fracture walls occurs. Both the normal Signorini-type/traction constraint and the Tresca friction law are recast as projection identities, and are combined into a single augmented Lagrangian bilinear form. The displacement is discretised by a DDR space of arbitrary polynomial degree k≥ 0, enriched on the fracture faces by bubble degrees of freedom that ensure stability via the existence of a discrete lifting between the Lagrange multiplier space and the discrete jump of the displacement along the fracture. We establish an abstract error estimate yielding optimal order convergence for the displacement and the Lagrange multiplier variables. Extensive numerical experiments confirm the theoretical analysis and demonstrate the practical accuracy and robustness of the scheme.
ER -