Finite Elements
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Related Articles from SNS
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....
A posteriori existence for the Keller-Segel model via a finite volume - finite element scheme
arXiv:2509.17710v2 Announce Type: replace Abstract: We derive two forms of conditional a posteriori error estimates for a finite volume scheme approximating the parabolic-elliptic Keller-Segel system. The estimates control the error in the $L^\infty(0,T, L^2(\Omega))$- and $L^2(0,T;H^1(\Omega))$-norm and exhibit linear convergence in the mesh size, as observed in numerical experiments. Crucially, we show that, as long as the condition of the error estimate is satisfied, a weak solution exists.
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...
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...
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....
Mesh Graph Neural Network Framework for Accelerating Finite Element Simulation for Arbitrary Geometries
arXiv:2606.08287v1 Announce Type: new Abstract: Finite element analysis (FEA) is essential for structural design but remains computationally expensive, particularly when evaluating multiple design iterations or load scenarios. Machine learning surrogate models offer a promising alternative, yet most approaches struggle with a critical limitation: generalizing across varying geometries. This work presents a mesh graph network (MGN) for predicting von Mises stress fields in 2D structural...
A Divergence-Free Scott-Vogelius Finite Element Method for the Surface Stokes Problem
arXiv:2606.07840v1 Announce Type: new Abstract: We construct and analyze an exactly divergence-free Scott-Vogelius finite element method for the surface Stokes problem. The proposed scheme simultaneously enforces the tangentiality and incompressibility constructs exactly and has the same number of unknowns as the two-dimensional Euclidean discretization.
Multicontinuum Generalized Multiscale Finite Element Method (MC-GMsFEM). Theory and applications to upscaling of two-phase flow
arXiv:2606.01303v1 Announce Type: new Abstract: We develop a multicontinuum Generalized Multiscale Finite Element Method (MC-GMsFEM) for constructing coarse-scale models in heterogeneous media that simultaneously provide accurate numerical approximations and physically consistent macroscopic equations. Classical multiscale methods efficiently approximate fine-scale solutions on coarse grids using localized basis functions, but they do not offer a systematic pathway for deriving macroscopic...
An energy-stable parametric finite element method for the Willmore flow in three dimensions
arXiv:2506.21025v3 Announce Type: replace Abstract: This work develops novel energy-stable parametric finite element methods (ES-PFEM) for the Willmore flow and curvature-dependent geometric gradient flows of surfaces in three dimensions. The key to achieving the energy stability lies in the use of two novel geometric identities: (i) a reformulated variational form of the normal velocity field, and (ii) incorporation of the temporal evolution of the mean curvature into the governing...