A two-level preconditioner for discontinuous virtual element methods with irregular subdomains
Abstract
We introduce and analyze a two-level overlapping additive Schwarz preconditioner for discontinuous Galerkin discretizations of virtual element spaces on general polygonal meshes in two dimensions, for a second-order elliptic problem whose coefficient is constant on each subdomain and may have large jumps across the interface. The discretization is a weighted symmetric interior penalty formulation: the averages carry the coefficient and the penalty is scaled by its harmonic mean across each edge, which keeps the method consistent and makes the constants of the analysis independent of the coefficient. The coarse space is obtained from a virtual, continuous partition of unity with a single degree of freedom per subdomain vertex, and is therefore applicable to the irregular subdomains produced by a graph partitioner; the discrete harmonic extension used to build it may be replaced by a cheaper projection onto polynomials of low degree. We prove that the condition number of the preconditioned system is bounded by C(1+log(H/h))(1+H/δ), where H, h and δ denote the subdomain diameter, the mesh size and the overlap, and where C is independent of the number of subdomains and of the contrast of the coefficient, provided the latter is quasi-monotone. This is the same bound known for the conforming case and the analysis allows for hanging nodes. We also show, by an explicit example, that the quasi-monotonicity assumption cannot be dispensed with. Numerical experiments with structured triangular, hexagonal and Voronoi meshes, and with subdomains obtained both by uniform partitioning and by METIS, confirm the theory.
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BibTeX
@misc{calvo2026twolevel,
title = {{A two-level preconditioner for discontinuous virtual element methods with irregular subdomains}},
author = {Juan G. Calvo and César Herrera},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18580},
eprint = {2610.18580},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18580},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A two-level preconditioner for discontinuous virtual element methods with irregular subdomains
AU - Juan G. Calvo
AU - César Herrera
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18580
UR - https://arxiv.org/abs/2610.18580
AB - We introduce and analyze a two-level overlapping additive Schwarz preconditioner for discontinuous Galerkin discretizations of virtual element spaces on general polygonal meshes in two dimensions, for a second-order elliptic problem whose coefficient is constant on each subdomain and may have large jumps across the interface. The discretization is a weighted symmetric interior penalty formulation: the averages carry the coefficient and the penalty is scaled by its harmonic mean across each edge, which keeps the method consistent and makes the constants of the analysis independent of the coefficient. The coarse space is obtained from a virtual, continuous partition of unity with a single degree of freedom per subdomain vertex, and is therefore applicable to the irregular subdomains produced by a graph partitioner; the discrete harmonic extension used to build it may be replaced by a cheaper projection onto polynomials of low degree. We prove that the condition number of the preconditioned system is bounded by C(1+log(H/h))(1+H/δ), where H, h and δ denote the subdomain diameter, the mesh size and the overlap, and where C is independent of the number of subdomains and of the contrast of the coefficient, provided the latter is quasi-monotone. This is the same bound known for the conforming case and the analysis allows for hanging nodes. We also show, by an explicit example, that the quasi-monotonicity assumption cannot be dispensed with. Numerical experiments with structured triangular, hexagonal and Voronoi meshes, and with subdomains obtained both by uniform partitioning and by METIS, confirm the theory.
ER -