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arXiv2026

Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines

We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines.

physics.flu-dyn↗

arXiv2026

ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients

Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.

math.NA↗

arXiv2026

Convergence of a continuous Galerkin method for the Biot-Allard poroelasticity system

We study a space-time finite element method for a system of poromechanics with memory effects that are modeled by a convolution integral. In the literature, the system is referred to as the Biot-Allard model. We recast the model as a first-order system in time, where the memory effects are transformed into an auxiliary differential equation. This allows for a computationally efficient numerical scheme. The system is discretized by continuous Galerkin methods in time and equal-order finite element methods in space. An optimal order error estimate is proved for the norm of the first-order energy of the unknowns of the system. The estimate is confirmed by numerical experiments.

math.NA↗

arXiv2026

Adaptive Multilevel Discontinuous Galerkin Methods on GPUs

I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary \emph{shadow cells} represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal--dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements ($p=3$) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.

math.NA↗

arXiv2026

Exact affine conditioning beyond Gaussians: a unique characterization of the ensemble Kalman update

The analysis step of the stochastic ensemble Kalman filter, called the ensemble Kalman update (EnKU), is widely used for approximating posterior distributions in inverse problems and data assimilation. The EnKU approximates the posterior distribution $π_{X\mid Y=y_\star}$ by pushing forward the joint distribution $(X,Y)\simπ$ through an affine map $L^{\mathrm{EnKU}}_{π,y_\star}(x,y)$ that depends only on the covariance structure of $π$ and the observation $y_\star$. While the EnKU yields the exact posterior for Gaussian $π$ in the mean-field, this property alone does not uniquely determine the EnKU. In fact, there are infinitely many affine maps $L_{π, y_\star}$ that achieve such exact conditioning. In this paper, we offer a novel characterization of the EnKU among all such affine maps. We first exhaustively characterize the set ${E}^{\mathrm{EnKU}}$ of joint distributions for which the EnKU yields exact conditioning, showing that it is much larger than the set of Gaussians. Next, we show that except for a small class of highly symmetric distributions within ${E}^{\mathrm{EnKU}}$, the EnKU is the {unique} exact affine conditioning map. Further, we characterize the largest possible set of distributions ${F}$ for which a distribution-dependent, weakly observation-dependent, affine map exists, a class of transports that naturally includes the EnKU. We show that ${F}={E}^{\mathrm{EnKU}}\cup{S}_{\mathrm{nl-dec}}$ with a small symmetry class ${S}_{\mathrm{nl-dec}}$, meaning that for affine conditioning beyond the Gaussian setting, the EnKU has an exact set that is essentially maximally large.

math.ST↗

arXiv2026

Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration

We study the semi-discrete quadratic Wasserstein energy. The energy is nonsmooth at collisions of sites. We prove local Lipschitz continuity on the full configuration space, together with global semiconcavity, coercivity, and dissipativity; show that every global minimizer is interior and collision free; and establish $C^2$ regularity on the collision-free configuration space. The gradient is expressed through the barycenters of the balanced Laguerre cells, while the Hessian is given by an explicit facet formula and satisfies a global one-sided bound. We also solve the one-dimensional problem explicitly in each ordering chamber and give a two-site example on the unit square with non-minimizing Lloyd fixed points. For $d\ge 2$, we then formulate an entropy-regularized relaxed control of the Langevin temperature. The controlled dynamics is strongly well posed, nonexplosive, and collision free. Its value function is a classical interior solution of the exploratory Hamilton-Jacobi-Bellman equation; the Laplacian of the value function is locally $C^1$, which yields a locally Lipschitz optimal temperature feedback. Independently of this optimal-control result, for every fixed Borel temperature rule bounded away from zero, and every sufficiently small step size, the associated Gaussian Euler chain is geometrically ergodic with a full-support invariant law. The raw iterates do not converge, whereas the best-so-far energy converges almost surely to the global minimum and the running record approaches the set of global minimizers.

math.NA↗

arXiv2026

Spectral Convergence of Random Feature Method in Multiple Dimensions

We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with $\tanh$ features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.

math.NA↗

arXiv2026

Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem

The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cosθ)^3$ where $e$ is the relative eccentricity of the primaries and $θ$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

math.OC↗

arXiv2026

Estimating systematic errors in Bayesian inversion using transport maps

In indirect measurements, the sought parameters have to be determined by solving an inverse problem, typically in a Bayesian framework. Often, the accurate numerical simulation of the measuring process is computationally demanding, making it necessary to rely on approximate models. These surrogates, however, introduce an additional model error and thus may distort the resulting parameter distribution. Moreover, even with the additional speed granted by the surrogate, posterior determination through conventional means such as Markov chain Monte Carlo might be cost intensive, specifically for complicated posterior shapes. In this paper, we propose a unified framework that combines Bayesian inference, model error correction and a transport-based sampling scheme to address these issues. To train the transport scheme, we investigate two different losses: one equivalent to the Kullback-Leibler divergence associated to the transport problem and one based on an upper bound of this loss, generally known as the evidence lower bound. We demonstrate that training the transport based on the latter changes the optimisation landscape drastically, potentially introducing an undesired bias in approximating the target posterior. We compare the computational cost of our approach with established methods and underline the theoretical results with numerical examples.

stat.ME↗

arXiv2026

A class of low-rank short recurrences for nonsymmetric linear matrix equations

We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. Building on ideas underpinning the recently introduced subspace conjugate gradient algorithm, we derive low-rank short recurrences that generalize one-dimensional subspace projection methods to the nonsymmetric matrix equation setting. To limit memory consumption and maximize computational efficiency, rank truncation strategies and modern randomization procedures are incorporated into the proposed algorithms. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.

math.NA↗

arXiv2026

Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential

This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.

math.NA↗

arXiv2026

A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws

We propose a novel framework for deriving semi-discrete discontinuous-Galerkin (DG) methods using operator semigroups for scalar conservation laws, then apply it to construct a class of high-order OFDG-type schemes [13] satisfying infinitely many local entropy inequalities with general E-fluxes on non-uniform meshes. Such schemes are further generalized to systems of conservation laws in any number of space dimensions by using entropy stable numerical fluxes in the sense of [1]. Finally, we prove optimal error estimates for smooth solutions to nonlinear scalar conservation laws, and prove strong convergence for discontinuous solutions to strictly convex conservation laws via compensated compactness.

math.NA↗

arXiv2026

A coercive space-time variational approach to fractional diffusion problems

We consider a fractional diffusion problem with temporal nonlocality acting on the diffusive flux. A coercive space--time variational formulation in Bochner-valued fractional Sobolev spaces is derived and the existence, uniqueness, and regularity of solutions are established. We further develop a conforming tensor-product Galerkin discretization and prove quasi-optimal error estimates in the anisotropic energy norm and improved convergence rates in weaker norms using duality arguments. In contrast to some space-time formulations for classical diffusion, the method preserves the causal structure of the evolution problem and leads to a time-stepping procedure with memory terms. On uniform time grids, the discrete history operator has a lower-triangular Toeplitz structure which enables an efficient implementation using fast recursive convolution techniques.

math.NA↗

arXiv2026

A stabilized scheme satisfying the discrete maximum principle for a time fractional convection-diffusion-reaction equation

We study a time fractional convection-diffusion-reaction equation in a bounded domain $Ω\subset\mathbb{R}^2$. A stabilized numerical scheme satisfying a discrete maximum principle is constructed by combining the conforming linear finite element method with the algebraic flux correction method. The resulting semi-discrete scheme is nonlinear, and its well-posedness is established. Assuming nonsmooth initial data, we derive error estimates for the semi-discrete scheme using energy arguments. For the temporal discretization, we employ the L1 method, obtaining a fully discrete scheme for which we prove well-posedness and the discrete maximum principle. We also present numerical experiments that validate the order of convergence as well as we test our schemes to solutions that possess layers.

math.NA↗

arXiv2026

Construction of trace-preserving Fortin operators

We present a unifying framework to construct local Fortin operators for conforming mixed finite element pairs for the Stokes equations. The operators are constructed to satisfy the divergence-preservation property, local stability and approximation properties, and certain trace-preservation properties. For the latter, some of the finite element pairs require an enrichment of the velocity space. We present the construction for the $P_d-P_0$ element, the Bernardi-Raugel element, the conforming Crouzeix-Raviart element, a modified MINI element and generalised Taylor-Hood elements. Furthermore, we discuss implications of the existence of such a Fortin operator beyond inf-sup stability. These include applications to non-Newtonian fluid flow, problems with inhomogeneous Dirichlet boundary conditions, as well as uniform inf-sup stability relevant for domain approximation and moving domains.

math.NA↗

arXiv2026

Random attractors and almost-sure stability under discretization of a stochastic autoparametric system

For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

math.DS↗

arXiv2026

Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability

We develop a reaction--diffusion--chemotaxis model for spatial tumour--immune--chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the nondimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies $σ_0/δ>1$, whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity $ξ_c$. A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results.

math.AP↗

arXiv2026

An Iterative Method with Asymptotic Orthogonality for Simultaneous Eigenpair Computation

The simultaneous computation of a cluster of eigenpairs with mutually orthogonal eigenvectors is a basic task in scientific computing. We develop a predictor--corrector discretization of the quasi-Grassmannian gradient flow for simultaneous eigenpair computation. The proposed iteration requires neither orthogonal initial data nor any orthogonalization operation: the predictor preserves the current Gram matrix, whereas the corrector reduces the orthogonality error, so that the iterates approach orthogonality asymptotically. We establish well-posedness of the discrete scheme and prove invariant-subspace nonexpansion, asymptotic orthogonality, energy decrease, and convergence to the target eigenspace. Numerical experiments with the discrete equations solved approximately show that the numerical iterates converge to eigenpair approximations whose eigenvectors are numerically orthogonal to high accuracy, while the energy and gradient norm decrease over the iterations.

math.NA↗