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Stabilization-free virtual element methods based on finite element interpolation
Announce Type: new Abstract: In this paper, we introduce a new framework for designing stabilization-free virtual element methods (VEMs) based on an finite element interpolation-based strategy, where we can simultaneously eliminate the stabilization terms in the discretizations of diffusion and reaction terms. The core idea is to construct a computable, polynomial-preserving, and norm-equivalent interpolation operator from the virtual element space to a (local) finite element space....
High-order conforming finite elements for the Cahn-Hilliard equation: Relative-energy stability and energy defects
arXiv:2606.06719v1 Announce Type: new Abstract: We study a semidiscrete single-field Galerkin approximation of the Cahn-Hilliard equation using high-order conforming finite element spaces. More specifically, globally $C^1$ finite elements with $H^2$-conforming trial spaces, including Argyris, Bell, and Bogner-Fox-Schmit elements, allow a direct discretization of the fourth-order formulation and preserve mass exactly. The main structural result is an exact energy balance for the physical...
A Mixed Extended Virtual Element Method for Elliptic Interface Problems on Polygonal Meshes
arXiv:2606.08526v1 Announce Type: new Abstract: We propose a lowest-order \(H(\operatorname{div})\)-conforming mixed extended virtual element method for elliptic interface problems on interface-unfitted polygonal meshes. The flux and pressure are approximated by subdomain-wise extended \(H(\operatorname{div})\)-VEM spaces and by piecewise constants, respectively. On cut elements, the computable polynomial projection is defined on the whole background element and then restricted to the two...
Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs
Announce Type: replace Abstract: In this paper, we study the finite element operator network (FEONet), an operator-learning method for parametric problems, originally introduced in J. Y. Lee, S. Ko, and Y. Hong, Finite Element Operator Network for Solving Elliptic-Type Parametric PDEs, SIAM J. Sci. Comput., 47(2), C501-C528, 2025. FEONet realizes the parameter-to-solution map on a finite element space and admits a training procedure that does not require training data, while exhibiting high...
The MINI mixed virtual element for the Stokes equation
Mathematics > Numerical Analysis [Submitted on 26 Mar 2025 (v1), last revised 7 Jun 2026 (this version, v2)] Title:The MINI mixed virtual element for the Stokes equation View PDF HTML (experimental)Abstract:We present and discuss a generalization of the popular MINI mixed finite element for the 2D Stokes equation by means of conforming virtual elements on polygonal meshes. We prove optimal error estimates for both velocity and pressure.
CREP: Cis-Regulatory Element Predictor Based on Fine-Tuned Enformer
A substantial fraction of disease-associated genetic variants reside in non-coding regions of the genome, where they act by perturbing cis-regulatory elements (CREs) such as enhancers, promoters, and insulators. While recent sequence-based deep learning models, such as Enformer, accurately predict continuous epigenomic signals from DNA sequence, they do not directly provide discrete and interpretable CRE annotations. Here, we present CREP (Cis-Regulatory Element Predictor), a fine-tuned...
Samsung Electronics invests $175 million in Element Biosciences' Series E funding round
Samsung Electronics invests $175 million in Element Biosciences' Series E funding round June 9 : Privately held Element Biosciences said on Tuesday it had raised funds in an upsized Series E round, including $175 million from longtime investor Samsung Electronics, to expand commercialization of its genetic testing and research products. The San Diego-based company did not disclose the total size of the round or its valuation. It also did not name the other investors.
Hessian-recovery-based C0 finite element methods for non-divergence form elliptic equations
arXiv:2606.03276v1 Announce Type: new Abstract: A Hessian-recovery-based C0 finite element framework is proposed for second-order elliptic equations in non-divergence form. The construction is based on a direct approximation of the strong non-divergence operator: the Hessian D2u is replaced by a recovered Hessian Hhuh, so that A : D2u is approximated by A : Hhuh. The resulting discretizations include a nodal formulation and a Galerkin-type formulation for general Lagrange finite element...
Cohomology of Finite Element Stokes Complexes on Alfeld Splits
arXiv:2605.31348v1 Announce Type: new Abstract: We show that the cohomology of the finite element Stokes complex consisting of piecewise polynomials spaces on an Alfeld split mesh from Fu, Guzm\'{a}n, & Neilan (2020, Math. Comp., 89, 1059--1091) is isomorphic to the cohomologies of the continuous Stokes and de Rham complexes. We also construct novel "minimal" conforming finite element complexes where the $H^1$-conforming space is the lowest-order space from Guzm\'{a}n & Neilan (2018, SIAM J....
A Mixed Virtual Element Method for the p-Laplace equation
arXiv:2606.07477v1 Announce Type: new Abstract: We introduce and analyze a mixed Virtual Element Method for the $p$-Laplace equation in a non-Hilbertian setting, covering the full range $p \in (1, \infty)$. The discrete framework combines standard mixed Virtual Element spaces with a novel non-linear stabilization term designed to mimic the power-law structure of the continuous operator. We establish discrete inf-sup stability under non-Hilbertian norms and rigorously prove the continuity and...