Year: 2026
Type: preprint
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 $L^2$-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 $\mathbb P_1$ 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.
Source: arXiv (Cornell University)
Authors: Burman, Erik, Hansbo, Peter, Larson, Mats G.
Institutions: University College London, Jönköping University, Umeå University
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Awards: Augmented Lagrangian Methods in Computational Mechanics, Computational methods for inverse problems subject to wave equations in heterogeneous media, Multiscale CutFEM for Mixed Dimensional Problems, G5139893163, G5933223730, G7008022610, Finite element methods for modern computing architectures and machine learning applications, Continuous finite element methods for under resolved turbulence in compressible flow, G8364456573
OpenAlex ID: W7212975210