A fully discrete LBRFD-IPDG method for linear fourth-order parabolic equations
Abstract
We propose a fully discrete method for linear fourth-order parabolic equations with Dirichlet boundary conditions, combining an implicit LBRFD multistep scheme in time with a mixed interior penalty discontinuous Galerkin (IPDG) method in space. The temporal discretization employs equispaced linear barycentric rational interpolants and incorporates a startup procedure. To facilitate the spatial discretization, the original problem is reformulated through an auxiliary variable. For certain parameter pairs (n,d), the LBRFD method is shown to be A(α)-stable and to possess a wider stability angle than the corresponding BDFp method of the same order. Stability and a priori error estimates are established via a G-energy technique and the discrete Grönwall lemma. The theoretical analysis yields a total L2 error estimate of order hk−1+τp, where p=d if n−d is even and p=d+1 if n−d is odd. The reduced spatial convergence rate is attributed to boundary contributions on ∂Ω. Despite this theoretical prediction, numerical experiments confirm the stability and demonstrate optimal convergence of order hk+1+τp.
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BibTeX
@misc{huang2026fully,
title = {{A fully discrete LBRFD-IPDG method for linear fourth-order parabolic equations}},
author = {Hongying Huang and Hong Zhang and Yanming Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.19250},
eprint = {2608.19250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19250},
}
RIS
TY - UNPB
TI - A fully discrete LBRFD-IPDG method for linear fourth-order parabolic equations
AU - Hongying Huang
AU - Hong Zhang
AU - Yanming Zhang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2608.19250
UR - https://arxiv.org/abs/2608.19250
AB - We propose a fully discrete method for linear fourth-order parabolic equations with Dirichlet boundary conditions, combining an implicit LBRFD multistep scheme in time with a mixed interior penalty discontinuous Galerkin (IPDG) method in space. The temporal discretization employs equispaced linear barycentric rational interpolants and incorporates a startup procedure. To facilitate the spatial discretization, the original problem is reformulated through an auxiliary variable. For certain parameter pairs (n,d), the LBRFD method is shown to be A(α)-stable and to possess a wider stability angle than the corresponding BDFp method of the same order. Stability and a priori error estimates are established via a G-energy technique and the discrete Grönwall lemma. The theoretical analysis yields a total L^2 error estimate of order h^k−1+τ^p, where p=d if n−d is even and p=d+1 if n−d is odd. The reduced spatial convergence rate is attributed to boundary contributions on ∂Ω. Despite this theoretical prediction, numerical experiments confirm the stability and demonstrate optimal convergence of order h^k+1+τ^p.
ER -