Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion
Abstract
We analyse fully discrete Streamline Diffusion finite element methods, also known as SUPG methods, for time-dependent first-order systems with skew-symmetric spatial operators. To the best of our knowledge, this is the first stability and a priori error analysis of strongly consistent Streamline Diffusion finite elements combined with linear multistep schemes of order greater than two for multidimensional first-order hyperbolic systems. Local-in-time L2-residual minimisation singles out δ=b0τ, where δ is the stabilisation parameter, τ the time step, and b0 the current-time coefficient in the method average. For this choice, each implicit linear system is a symmetric positive definite normal equation in a graph norm. A direct dual-residual estimate yields errors of order O(hk+1/2+τν) when δ is proportional to h and τ=O(h), where h is the mesh size, k the polynomial degree, and ν the temporal order. Stability is established for the θ-method and the Adams-Moulton methods AM3-AM5. The general-degree AM3 and AM4 estimates use strengthened CFL conditions; for continuous piecewise affine elements, an elementwise cancellation recovers the standard hyperbolic regime. AM5 and the explicit Adams-Bashforth methods AB3 and AB4 are stable under a standard hyperbolic CFL condition, though the explicit schemes do not have normal-equation structure. Order-matched acoustic tests recover orders two through five and material-residual rates one half-order lower. In a d'Alembert test with discontinuous initial data, normal-equation SUPG recovers second-order local ℙ1 convergence away from the wave fronts, whereas the unstabilised error is close to half order. A two-dimensional compact-wave test shows improved localisation with modest dissipation.
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BibTeX
@misc{burman2026normal,
title = {{Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion}},
author = {Erik Burman and Peter Hansbo and Mats G. Larson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12506},
eprint = {2609.12506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12506},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Normal Equations and Discrete Energy Structures for High-Order Streamline Diffusion
AU - Erik Burman
AU - Peter Hansbo
AU - Mats G. Larson
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12506
UR - https://arxiv.org/abs/2609.12506
AB - We analyse fully discrete Streamline Diffusion finite element methods, also known as SUPG methods, for time-dependent first-order systems with skew-symmetric spatial operators. To the best of our knowledge, this is the first stability and a priori error analysis of strongly consistent Streamline Diffusion finite elements combined with linear multistep schemes of order greater than two for multidimensional first-order hyperbolic systems. Local-in-time L^2-residual minimisation singles out δ=b_0τ, where δ is the stabilisation parameter, τ the time step, and b_0 the current-time coefficient in the method average. For this choice, each implicit linear system is a symmetric positive definite normal equation in a graph norm. A direct dual-residual estimate yields errors of order O(h^k+1/2+τ^ν) when δ is proportional to h and τ=O(h), where h is the mesh size, k the polynomial degree, and ν the temporal order. Stability is established for the θ-method and the Adams-Moulton methods AM3-AM5. The general-degree AM3 and AM4 estimates use strengthened CFL conditions; for continuous piecewise affine elements, an elementwise cancellation recovers the standard hyperbolic regime. AM5 and the explicit Adams-Bashforth methods AB3 and AB4 are stable under a standard hyperbolic CFL condition, though the explicit schemes do not have normal-equation structure. Order-matched acoustic tests recover orders two through five and material-residual rates one half-order lower. In a d'Alembert test with discontinuous initial data, normal-equation SUPG recovers second-order local ℙ_1 convergence away from the wave fronts, whereas the unstabilised error is close to half order. A two-dimensional compact-wave test shows improved localisation with modest dissipation.
ER -