An Optimal IPDG Scheme for the Biharmonic Equation
Abstract
This paper presents an optimal interior penalty discontinuous Galerkin (IPDG) scheme for the planar biharmonic equation using piecewise polynomials of degree k=3 or $4$. In standard IPDG methods, large penalty parameters force the discrete solution into an overconstrained space, severely degrading accuracy---phenomenon known as numerical locking. To overcome this, our method enforces only vertex continuity and projects the jumps of the function and its normal derivative onto 𝒫k−3 and 𝒫k−2, respectively. Consequently, as the penalty parameters tend to infinity, the discrete solution is forced to lie in a constrained subspace Vh,∞k, which we identify as the optimal nonconforming finite element space Bhk. This intrinsic connection fundamentally eliminates numerical locking. We prove optimal error estimates of 𝒪(hk−1) in a mesh-dependent energy norm and 𝒪(hk+1) in the L2 norm. Numerical experiments on both convex and L-shaped domains confirm that the proposed scheme is robust and entirely locking-free, even for extremely large penalty parameters.
Related
- Existence of solutions for a nonlinear elliptic equation involving (p(x),q(x))-Laplacian and (p(x),q(x))-biharmonic Operator
- Existence Results for the Critical Biharmonic Elliptic Equation Involving Singular Nonlinearities
- Long-time reduction for the biharmonic nonlinear Schrödinger equation
- Failure of superharmonicity near an isolated singularity of the critical biharmonic equation
- Local well-posedness for the biharmonic nonlinear Schrödinger equation in low dimensions
BibTeX
@misc{zhang2026optimal,
title = {{An Optimal IPDG Scheme for the Biharmonic Equation}},
author = {Bohua Zhang and Xia Ji and Shuo Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12675},
eprint = {2609.12675},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12675},
}
RIS
TY - UNPB
TI - An Optimal IPDG Scheme for the Biharmonic Equation
AU - Bohua Zhang
AU - Xia Ji
AU - Shuo Zhang
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12675
UR - https://arxiv.org/abs/2609.12675
AB - This paper presents an optimal interior penalty discontinuous Galerkin (IPDG) scheme for the planar biharmonic equation using piecewise polynomials of degree k=3 or $4$. In standard IPDG methods, large penalty parameters force the discrete solution into an overconstrained space, severely degrading accuracy---phenomenon known as numerical locking. To overcome this, our method enforces only vertex continuity and projects the jumps of the function and its normal derivative onto 𝒫^k−3 and 𝒫^k−2, respectively. Consequently, as the penalty parameters tend to infinity, the discrete solution is forced to lie in a constrained subspace V_h,∞^k, which we identify as the optimal nonconforming finite element space B_h^k. This intrinsic connection fundamentally eliminates numerical locking. We prove optimal error estimates of 𝒪(h^k−1) in a mesh-dependent energy norm and 𝒪(h^k+1) in the L^2 norm. Numerical experiments on both convex and L-shaped domains confirm that the proposed scheme is robust and entirely locking-free, even for extremely large penalty parameters.
ER -