Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.08765

A finite element continuous data assimilation framework for a Navier--Stokes--Cahn--Hilliard system

Tianyu Sun
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper studies a coupled two-dimensional Navier--Stokes--Cahn--Hilliard phase-field model augmented by a transported auxiliary field, and develops a continuous data assimilation (CDA) framework for recovering its trajectories from coarse-in-space observations. We formulate a nudging-based CDA system for the coupled NSCH--auxiliary-field model, in which coarse measurements are incorporated through a general linear observation operator. The observation mechanism is described abstractly by an interpolant satisfying an $H^{2}$-type approximation property, which is compatible with coarse spatial observations obtained from mesh coarsening and reconstruction. At the continuous level, we record two structural properties of the model: a formal energy law for the reference system and an evolution law for the phase mean in the assimilated dynamics. At the discrete level, we introduce a capped fully discrete finite element splitting scheme using continuous quadratic elements for the phase, chemical potential, velocity, and auxiliary field, together with continuous linear elements for the pressure. For this scheme, we prove one-step well-posedness and establish a stepwise a priori estimate for the capped method. Numerical experiments illustrate the practical behavior of the proposed CDA approach. They demonstrate recovery from strongly mismatched initial conditions and show how synchronization depends on the observation resolution, boundary forcing, and feedback strength. A coarse-indistinguishability experiment further shows that identical coarse initial information may correspond to distinct fine-scale evolutions, while the assimilated dynamics select the trajectory determined by the supplied time-dependent observations.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tianyu Sun. 2026-03-09. A finite element continuous data assimilation framework for a Navier--Stokes--Cahn--Hilliard system. https://arxiv.org/abs/2603.08765

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

KEEP EXPLORING

Related papers

arXiv2026

How many continuous measurements are needed to learn a vector?

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.

math.NA↗
arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗