A robust finite element method for incompressible magnetohydrodynamics using Whitney forms

Tianwei Yu

Home/Archive/arXiv

arXiv · 2610.26606

A robust finite element method for incompressible magnetohydrodynamics using Whitney forms

Tianwei Yu

View original source ↗Read source document ↗Export metadata ↓

Abstract

A stabilized semi-discrete finite element scheme for the incompressible magnetohydrodynamics system is presented. The scheme uses the Whitney $2$-form and $1$-form for the fluid velocity and magnetic induction, respectively. The key ingredient is a magnetic curl-jump penalty term, which allows us to prove a pre-asymptotic error estimate of order $O(h^{1/2})$ with a constant depending on solution norms but not explicitly on the fluid viscosity and magnetic diffusivity.

Explore related subjects

math.NA cs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tianwei Yu. 2026-10-19. A robust finite element method for incompressible magnetohydrodynamics using Whitney forms. https://arxiv.org/abs/2610.26606

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

KEEP EXPLORING

Related papers

arXiv2026

Kernel-based collocation methods for Zakai equations

We analyze kernel-based interpolation methods for numerical solutions of Zakai equations in nonlinear filtering. The stability of the spatial discretization is quantified by row sums of the discrete differential operators. The resulting mean-square error estimate separates the one-step consistency error from its finite-time amplification and yields convergence when these two quantities balance. A short finite-resolution example illustrates the performance of a postprocessed interpolant.

math.NA↗

arXiv2026

Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions

We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal. 2020), in which a countable number of independent random variables enter the random field as periodic functions, it was shown by Kaarnioja, Kazashi, Kuo, Nobile, and Sloan (Numer. Math. 2022) that the lattice-based kernel interpolant can be constructed for the PDE solution as a function of the stochastic variables in a highly efficient manner using fast Fourier transform (FFT). In this work, we discuss the connection between our model and the popular "affine and uniform model" studied widely in the literature of uncertainty quantification for PDEs with uncertain coefficients. We also propose a new class of weights entering the construction of the kernel interpolant $-$ serendipitous weights $-$ which dramatically improve the computational performance of the kernel interpolant for PDE problems with uncertain coefficients, and allow us to tackle function approximation problems up to very high dimensionalities. Numerical experiments are presented to showcase the performance of the serendipitous weights.

math.NA↗

arXiv2026

Enhanced randomized Douglas-Rachford method for linear systems: Improved probabilities and adaptive momentum

Randomized iterative methods have gained recent interest in machine learning and signal processing for solving large-scale linear systems. One such example is the randomized Douglas-Rachford (RDR) method, which updates the iterate by reflecting it through two randomly selected hyperplanes and taking a convex combination with the current point. In this work, we enhance RDR by introducing improved sampling strategies and an adaptive heavy-ball momentum scheme. Specifically, we incorporate without-replacement sampling and volume sampling into RDR and establish linear convergence in expectation for both schemes. For without-replacement sampling, we further establish a larger one-step descent than that of conventional RDR. For volume sampling, the convergence rate explicitly incorporates geometric information, and a tighter comparison with RDR is obtained under additional structural assumptions. Furthermore, we develop an adaptive momentum mechanism that determines the relaxation and momentum parameters directly from the iteration information without manual tuning. We establish linear convergence in expectation for the resulting method. Numerical experiments demonstrate the effectiveness of the proposed methods across a range of problem settings.

math.NA↗