Jai Tushar: Thursday, 10th July 2025 at 10:30
Abstract:
Polytopal methods are a class of Finite Element Methods (FEMs) that have gained popularity in recent years due to their ability to relax conformity constraints on meshes. This flexibility makes them well-suited for handling complex geometries, adaptive mesh refinement
and coarsening. The design of efficient, robust, scalable solvers for linear systems arising from these kind of discretisations is important to make them competitive with traditional methods. One family of such scalable preconditioners are non-overlapping domain decomposition
methods.
The analysis of non-overlapping domain decomposition method-based solvers like BDDC relies on the exchange of information across inter-subdomain boundaries. It requires three main ingredients: a trace inequality, which implies that the restriction of functions to the
subdomain interface is stable; a lifting result, which lifts this restriction to the interior of the neighboring subdomain; and continuity of a face truncation operator on piecewise polynomial functions. The bound on this operator leads to a mesh-dependent logarithmic estimate.
For conforming finite element methods, this is realized with the help of continuous trace theory. For non-conforming methods, such as polytopal methods, the continuous trace theory fails, since the trace of piecewise polynomial functions in L2(Ω) does not possess H1/2(∂Ω)
regularity. The current state of the art to address this involves constructing an interpolant of a function on the interface (inter-subdomain boundary) onto a conforming finite element
space and then applying the continuous trace theory. As a result, all the analysis for nonconforming spaces so far has been carried out on conforming simplicial/tetrahedral or quadrilateral/hexahedral meshes.
In this talk, we will present a discrete trace theory for non-conforming polytopal methods. This theory is based entirely on the fully discrete hybrid spaces appearing in these methods. It hinges on the design of a novel discrete trace seminorm. For this seminorm, we establish discrete trace and lifting inequalities that are independent of mesh size and hold on quasi-uniform polytopal meshes. We also derive a truncation estimate in this discrete trace seminorm for piecewise polynomials in a hybrid setting. We will use these ingredients to prove condition number estimates for the Balancing Domain Decomposition by Constraints (BDDC) preconditioner. Then, we compute the proposed discrete trace operator and show that its spectrum is equivalent to that of the discrete H1-like operator (a consequence of the discrete trace and lifting inequalities). Finally, we show robustness and scalability of our preconditioner for up to several hundreds of processors on both structured and fully polytopal meshes, in 2D and 3D for the HDG and HHO schemes. All experiments are performed using the open-source finite element library Gridap.jl, and run on the GADI supercomputer using resources provided by the Australian Government through NCI under the NCMAS Merit Allocation Scheme. This talk is based on [1, 2].
References
[1] S. Badia, J. Droniou, and J. Tushar, “A discrete trace theory for non-conforming polytopal hybrid discretisation methods,” arXiv:2409.15863, 2024.
[2] S. Badia, J. Droniou, J. Manyer, and J. Tushar, “Analysis of BDDC preconditioners for nonconforming polytopal hybrid discretisation methods,” arXiv:2506.11956, 2025.
