arXiv subjects

Tim van Beeck

Publications and source records attributed to Tim van Beeck.

8 recordsLinked to original sources

arXiv2026

A hybrid $C^{0}$-interior penalty method for the nematic Helmholtz--Korteweg equation

The nematic Helmholtz--Korteweg equation models the propagation of time-harmonic acoustic waves in nematic Korteweg fluids, such as nematic liquid crystals. The PDE augments the classical Helmholtz equation with two additional fourth-order terms, one of which is anisotropic in the direction of the nematic field. We refine the previous continuous analysis of Farrell et al. (2025) by using the Cordes condition and present a $C^0$-hybrid interior penalty discretization. The proposed discretization offers greater flexibility than $C^1$-conforming methods and is well-suited for applications in three dimensions and on curved domains. We prove stability of the method for any polynomial degree greater than or equal to two, independent of the spatial dimension, provided that the anisotropy is sufficiently small. Further, we show that the sequence of discrete solutions converges to the continuous solution under minimal regularity assumptions and derive convergence rates if the continuous solution has additional regularity. Finally, we illustrate the capabilities of the method through numerical examples.

math.NA↗

arXiv2026

Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions

We present a discrete Helmholtz--Hodge decomposition for H(div)-conforming Brezzi--Douglas--Marini (BDM) finite elements on triangulated surfaces of arbitrary topology. The divergence-free BDM subspace is split L2-orthogonally into rotated gradients of a continuous streamfunction space and a finite-dimensional space of discrete harmonic fields whose dimension equals the first Betti number of the surface. Consequently, any incompressible flow discretized on this subspace can be reformulated with a scalar streamfunction and finitely many harmonic coefficients as the only unknowns. This eliminates the pressure and the saddle-point structure while ensuring exact tangentiality, pointwise divergence-freeness, and pressure-robustness. We present a randomized algorithm for constructing the harmonic basis and discuss implementation aspects including hybridization, efficient treatment of the harmonic unknowns, and pressure reconstruction. Numerical experiments for unsteady surface Navier--Stokes equations on a trefoil knot and a multiply-connected sculpture surface demonstrate the method and illustrate the physical role of the harmonic velocity component.

math.NA↗

arXiv2026

Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction

This paper studies pressure-robustness for the axisymmetric Stokes problem. The transformation to cylindrical coordinates requires that the radially weighted velocity is divergence-free in the classical sense. Consequently, traditional divergence-free finite element methods from the Cartesian setting -- even if inf-sup stable -- are in general not divergence-free in the axisymmetric formulation. We therefore explore the approach that restores pressure-robustness via reconstruction operators for a low-order Bernardi--Raugel discretization. We show that an application of standard interpolation operators from the Cartesian setting to radially weighted test functions works in principle, but it lacks properties needed to derive optimal consistency error estimates. To address this, we introduce a reconstruction operator into a finite element space spanned by Raviart--Thomas functions that are modified such that they vanish on the rotation axis. This vanishing-on-axis property is the key to obtain optimal consistency error estimates. Numerical examples demonstrate the overall feasibility of the approach and include cases where the vanishing-on-axis property yields significantly better results.

math.NA↗

arXiv2025

Variational data assimilation for the wave equation in heterogeneous media: Numerical investigation of stability

In recent years, several numerical methods for solving the unique continuation problem for the wave equation in a homogeneous medium with given data on the lateral boundary of the space-time cylinder have been proposed. This problem enjoys Lipschitz stability if the geometric control condition is fulfilled, which allows devising optimally convergent numerical methods. In this article, we investigate whether these results carry over to the case in which the medium exhibits a jump discontinuity. Our numerical experiments suggest a positive answer. However, we also observe that the presence of discontinuities in the medium renders the computations far more demanding than in the homogeneous case.

math.NA↗

arXiv2025

Hybrid discontinuous Galerkin discretizations for the damped time-harmonic Galbrun's equation

In this article, we study the damped time-harmonic Galbrun's equation which models solar and stellar oscillations. We introduce and analyze hybrid discontinuous Galerkin discretizations (HDG) that are stable and optimally convergent for all polynomial degrees greater than or equal to one. The proposed methods are robust with respect to the drastic changes in the magnitude of the coefficients that naturally occur in stars. Our analysis is based on the concept of discrete approximation schemes and weak T-compatibility, which exploits the weakly T-coercive structure of the equation. Compared to the $H^1$-conforming discretization of [Halla, Lehrenfeld, Stocker, 2022], our method offers improved stability and robustness. Furthermore, it significantly reduces the computational costs compared to the $H(\operatorname{div})$-conforming DG discretization of [Halla, 2023], which has similar stability properties. These advantages make the proposed HDG methods well-suited for astrophysical simulations.

math.NA↗

arXiv2025

Analysis and numerical analysis of the Helmholtz-Korteweg equation

We analyse the nematic Helmholtz-Korteweg equation, a variant of the classical Helmholtz equation that describes time-harmonic wave propagation in calamitic fluids in the presence of nematic order. A prominent example is given by nematic liquid crystals, which can be modeled as nematic Korteweg fluids - that is, fluids whose stress tensor depends on density gradients and on a nematic director describing the orientation of the anisotropic molecules. These materials exhibit anisotropic acoustic properties that can be tuned by external electromagnetic fields, making them attractive for potential applications such as tunable acoustic resonators. We prove the existence and uniqueness of solutions to this equation in two and three dimensions for suitable (nonresonant) wave numbers and propose a convergent discretisation for its numerical solution. The discretisation of this problem is nontrivial as it demands high regularity and involves unfamiliar boundary conditions. We address these challenges by using high-order conforming finite elements and enforcing the boundary conditions with Nitsche's method. We illustrate our analysis with numerical simulations in two dimensions.

math.NA↗

arXiv2024

An adaptive mesh refinement strategy to ensure quasi-optimality of finite element methods for self-adjoint Helmholtz problems

It is well known that the quasi-optimality of the Galerkin finite element method for the Helmholtz equation is dependent on the mesh size and the wave-number. In the literature, different criteria have been proposed to ensure uniform quasi-optimality of the discretisation. In the present work, we study the uniform quasi-optimality of $H^1$ conforming and non-conforming Crouzeix-Raviart discretisation of the self-adjoint Helmholtz problem. In particular, we propose an adaptive scheme, coupled with a residual-based indicator, for generating guaranteed quasi-optimal meshes with minimal degrees of freedom.

math.NA↗

arXiv2023

Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem

In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.

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