A Double-Adaptivity Solver for Parabolic PDEs
Abstract
We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation -- being the secondary variable of our system -- under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in 1+1 and 2+1 dimensions.
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BibTeX
@misc{gantner2026doubleadaptivity,
title = {{A Double-Adaptivity Solver for Parabolic PDEs}},
author = {Gregor Gantner and Robin Smeets and Rob Stevenson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01748},
eprint = {2609.01748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01748},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Double-Adaptivity Solver for Parabolic PDEs
AU - Gregor Gantner
AU - Robin Smeets
AU - Rob Stevenson
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01748
UR - https://arxiv.org/abs/2609.01748
AB - We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation -- being the secondary variable of our system -- under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in 1+1 and 2+1 dimensions.
ER -