Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39325

A priori error analysis for the Shliomis ferrofluid model without magnetic regularization

Gonzalo A. Benavides·Ricardo H. Nochetto
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop and analyze a family of conforming, energy-stable spatial semidiscretizations of the Shliomis ferrofluid model in two and three space dimensions. Unlike many existing numerical analyses, we treat the system without introducing magnetic diffusion. The limited spatial control supplied by the energy law complicates the treatment of magnetic nonlinearities in the error analysis. Our formulation retains the rotational coupling and nonlinear magnetic relaxation and incorporates the magnetic normal-flux condition naturally, overcoming the dimensional restriction and boundary-enforcement difficulties of the two-potential approach of Benavides, Nochetto and Shakipov (2026). The error analysis combines consistent stabilization of the Maxwell constraint with a cancellation arising from exact incompressibility and the pointwise symmetry of the Hessian of the magnetic potential. We prove global solvability of the semidiscrete scheme and establish a priori error estimates for sufficiently regular exact solutions under explicit approximation and projection-stability assumptions. We also construct an averaged $\mathrm{H}^2$-conforming projector, following Girault and Scott (2002), that is defined on $\mathrm{L}^1(Ω)$, and establish its Sobolev approximation and stability properties. These properties are needed for the error analysis and are of independent interest.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gonzalo A. Benavides, Ricardo H. Nochetto. 2026-10-29. A priori error analysis for the Shliomis ferrofluid model without magnetic regularization. https://arxiv.org/abs/2610.39325

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A continuous scale space of diffeomorphisms

In this paper, we define and study a nested family of reproducing kernel Hilbert spaces of vector fields that is indexed by a range of scales, from which we construct a reproducing kernel Hilbert space of scale-dependent vector fields. We provide a characterization of the reproducing kernel of that space, with numerical approximations ensuring quick evaluations when this kernel does not have a closed form. We then introduce a multiscale version of the large deformation diffeomorphic metric mapping (LDDMM) problem and prove the existence of solutions. Finally, we provide numerical experiments performing landmark matching using multiscale LDDMM.

math.NA↗
arXiv2026

Hierarchical Matrices with Tucker Blocks: Construction and Forward Evaluation

In this paper, we introduce hierarchical matrices with Tucker blocks (HMTB) for data-sparse approximation of matrices arising from nonoscillatory and mildly oscillatory kernels. Building on the hierarchical matrix framework, HMTB matrices replace conventional low-rank blocks with Tucker representations. Using high-dimensional interpolation and tensor compression and contractions, we develop algorithms for constructing HMTB matrices and performing matrix-vector multiplication. For an $N \times N$ matrix, these algorithms have linear and quasi-linear complexities in $N$, respectively. Numerical comparisons with an interpolation-based $\mathcal{H}$-matrix implementation demonstrate storage savings and, in several cases, reduced construction times, while maintaining competitive forward-evaluation costs. The framework is also applicable to quasi-uniform grids.

math.NA↗
arXiv2026

Data-driven model order reduction for wave propagation in materials with damage and nonlinearities

In this work, we consider wave propagation in materials characterized by nonlinear properties or damage. To accelerate the simulations of the resulting high-dimensional problems, we apply model order reduction methods. Depending on the knowledge of the underlying equations and the availability of their discrete operators, intrusive methods (here projection-based approaches based on proper orthogonal decomposition (POD)) or non-instrusive methods (here data-driven approaches including dynamic mode decomposition (DMD) and operator inference (OpInf)) can be used. We recall the theoretical foundations of the methods and apply them to the problem of wave propagation. In three different numerical examples, we evaluate the performance of the reduction techniques.

math.NA↗