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arXiv2026

Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(Ω)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.

math.NA↗

arXiv2026

Scalable Pseudospectral Analysis via Low-Rank Approximations of Dynamical Systems

Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While existing research on scalable pseudospectral computation has focused on exploiting sparsity structures, common in discretizations of differential operators, these approaches are ill-suited for machine learning and data-driven dynamical systems, where operators are typically dense but approximately low-rank. In this paper, we develop a comprehensive low-rank framework that dramatically reduces this computational burden. Our core theoretical contribution is an exact characterization of the pseudospectrum of arbitrary low-rank matrices, reducing the evaluation of resolvent norms to eigenvalue problems of dimension proportional to the rank. Building on this foundation, we derive rigorous inclusion sets for the pseudospectra of general matrices via truncated and randomized low-rank approximations, with explicit perturbation bounds. These results enable efficient estimators for key stability quantities, including distance to instability and Kreiss constants, at a cost that scales with the effective rank rather than the ambient dimension. We further demonstrate how our framework naturally extends to data-driven settings, providing pseudospectral analysis of transfer operators learned from nonlinear and stochastic dynamical systems. Numerical experiments confirm orders-of-magnitude speedups while preserving accuracy, opening pseudospectral analysis to previously intractable high-dimensional problems in computational PDEs, control theory, and data-driven dynamics.

math.NA↗

arXiv2026

A non-intrusive approach to index-aware learning

We present a non-intrusive version of the index-aware learning framework introduced in arXiv:2309.00958. Index-aware learning itself is an approach for learning the time and parameter dependent solutions of differential-algebraic equations (DAEs), in particular those describing electric circuits. A central feature of the approach is that it ensures the learned solutions to fulfill the inherent constraints of the DAE, such as e.g. Kirchhoff's laws in the case of electric circuits. This is achieved by leveraging a decoupling of the DAE into its differential and algebraic parts, with the non-intrusive version of the approach additionally relying on results from arXiv:2604.20475 and arXiv:2107.07755. We illustrate the approach using a filtered buck converter as an example and compare both the intrusive and non-intrusive versions. The code for the example is openly available.

cs.CE↗

arXiv2026

Mamba-Assisted Non-Markovian Closure for Reduced-Order Modeling

Reduced-order modeling of high-dimensional dynamical systems is often hindered by closure effects arising from unresolved variables, which can introduce non-Markovian dependence into the resolved dynamics. Motivated by the history-dependent memory term arising in the Mori--Zwanzig formalism, we recast non-Markovian closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework. MAC employs a Mamba-based sequence model to predict the closure from the resolved trajectory and couples the learned closure with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. During training, the selective scan mechanism in Mamba enables efficient parallel sequence processing with linear scaling in sequence length, while autoregressive inference proceeds through recurrent state updates at essentially constant per-step cost. We evaluate MAC on four benchmark systems with complementary characteristics: the viscous Burgers' equation, the chaotic two-scale Lorenz '96 system, the 3-bus DeMarco--Zheng power-grid system, and the dispersive Korteweg--de Vries equation. Across these benchmarks, MAC consistently improves predictive accuracy and long-time rollout stability relative to the comparison models, demonstrating an effective and computationally scalable approach to non-Markovian closure modeling.

cs.LG↗

arXiv2026

An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions

For wave scattering by axisymmetric surfaces, the fast Fourier transform (FFT) method provides an effective tool to accelerate standard boundary integral equation (BIE) solvers. Surface BIEs can be decoupled into a series of curve integral equations on the generating curve, due to the convolution-like integral operators. The Fourier coefficients of the three-dimensional fundamental kernels can be rapidly computed through three-term recurrence relations based on Miller's algorithm. Such well-established techniques break down for nonaxisymmetric surfaces. This paper proposes a novel FFT-accelerated boundary integral method for wave scattering by smooth surfaces of arbitrary shapes. The Fourier coefficients of the singular kernels now satisfy higher-order recurrence relations. Although they can be solved with an optimal linear complexity by the standard Olver's algorithm, it turns out that a singularity swapping approach that rewrites each kernel as the product of a smooth function and an axisymmetric-related singular factor is realistically much faster. Consequently, Miller's algorithm together with the standard FFT convolution yields an ${\cal O}(M\log M)$ approach for evaluating the ${\cal O}(M)$ Fourier coefficients of the kernels, attaining exactly the same order of complexity for axisymmetric surfaces! With such FFT-based efficient procedures, we rewrite the surface BIEs in terms of ${\cal O}(M)$ curve integrals, which are proved to exhibit logarithmic singularities, discretize them by panel-based generalized Gaussian quadratures, and obtain spectrally accurate linear systems to approximate the wavefields. Extensive numerical experiments are carried out to demonstrate the effectiveness and spectral accuracy of the new approach.

math.NA↗

arXiv2026

Entry growth in Gaussian elimination

Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s. Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. In contrast, we show that every non-singular matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard.

math.NA↗

arXiv2026

Waveguiding in systems of high contrast resonators: Theory and fast computations

In this work, we study guided modes in systems of high-contrast resonators near nonzero interior Neumann frequencies, beyond the subwavelength regime. In the regular exterior regime, where the exterior Dirichlet problem is well-posed at the reference wavenumber, we introduce an infinite-dimensional frequency-dependent capacitance operator obtained by compressing the exterior Helmholtz Dirichlet-to-Neumann map to the traces of the interior resonant Neumann eigenspaces. We prove the norm-resolvent convergence of the continuous problem to this discrete effective operator as the contrast $δ\to0$, and derive first-order asymptotic formulas for compact-defect frequencies and line-defect band functions. We then establish exponential off-diagonal decay of the capacitance coefficients by a Combes--Thomas argument, yielding an exponentially accurate truncation of the discrete operator, and show that its retained coefficients can be computed from local Helmholtz problems. At the physical frequency, this local approximation converges exponentially under a uniform stability assumption for the growing finite-cluster problems. The stability assumption can be removed by introducing a vanishing complex absorption together with a Hermitian symmetrization. In particular, an absorption parameter of order $\sqrtδ$, together with interaction truncation and patch radii of order $|\logδ|$, suffices to preserve the $O(δ^2)$ accuracy of the first-order high-contrast expansion of the defect eigenfrequencies, yielding a fast computational method. Numerical experiments for dipole and quadrupole resonances illustrate the accuracy, exponential locality, and applicability of the discrete model to straight and bent waveguides generated by material or geometric detuning.

math.NA↗

arXiv2026

Structure-Preserving Detailed-Balance Master-Equation Discretizations for Fokker--Planck Equations

We develop a variational--Markov construction of detailed-balance master-equation discretizations for Fokker--Planck equations directly from their energy--dissipation laws. Rather than discretizing the differential operator, we represent mass transfer between neighboring grid points by two directional jump rates. On each edge, the discrete energy--dissipation law determines the net flux, while detailed balance fixes the rate ratio; a local reconstruction of edge quantities from neighboring grid-point values then uniquely determines both rates. A logarithmic-mean reconstruction recovers the classical Scharfetter--Gummel/Wang--Peskin--Elston rates, while alternative reconstructions yield other reversible schemes. The resulting semi-discrete systems conserve mass, preserve nonnegativity, satisfy detailed balance, and dissipate a discrete free energy. The same edgewise construction extends to state-dependent and degenerate mobilities, nonlocal interaction energies and higher-dimensional problems. The corresponding implicit and semi-implicit fully discrete schemes are linear, mass-conservative, positivity-preserving, and energy-stable, with a time-step restriction required only for nonlocal interaction energies whose kernel is not negative semidefinite. Numerical experiments confirm the expected spatial convergence rates of all detailed-balance schemes and verify their equilibrium accuracy, positivity preservation, and free-energy decay with discontinuous potentials, saturation, nonlocal interactions, and two-dimensional problems.

math.NA↗

arXiv2026

Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator

For $N\geq 3$ and a potential phase $\vartheta\in\mathbb{R}$, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle $φ\in\mathbb{R}$, $A_N(φ,\vartheta)=e^{iφ}(S+S^{-1})/2+\operatorname{diag}(\cos(2πj/N+\vartheta))_{j\in\mathbb{Z}/N\mathbb{Z}}$, where $S e_j=e_{j+1}$ is the periodic shift on $\mathbb{C}^N$. We derive a Chambers formula and isolate the part $Q_{N,φ}$ of the characteristic polynomial that depends only on $N$ and $φ$, but not on $\vartheta$ or on a change of boundary conditions for the shift operator. We then show, for every $N$, that the zeros of $Q_{N,φ}$ lie on the two perpendicular lines $e^{iφ/2}\mathbb{R}\cup e^{i(φ/2+π/2)}\mathbb{R}$. For even $N$, the same property holds for the matrices $A_N(φ,\vartheta)$ with $\vartheta\in 2π\mathbb{Z}/N$, and we compute their limiting eigenvalue measure explicitly. For $φ\in[-π,π]$, the eigenvalue distribution approximates elliptic-integral densities with masses $1-|φ|/π$ and $|φ|/π$, and maximal radii $2|\cos(φ/2)|$ and $2|\sin(φ/2)|$, respectively. At $φ=π/2$, the central polynomial $Q_{N,φ}$ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.

math.SP↗

arXiv2026

Checkerboard Shells: A Position-Only Thin-Shell Discretization with Scale-Compatible Completion

Checkerboard edge-midpoint geometry provides an exact planar Varignon parallelogram for every spatial quadrilateral, allowing a local tangent frame and normal to be recovered directly from nodal positions even when the raw quadrilateral is warped. Building on this property, we develop a position-only thin-shell discretization with no independent director, rotation, or strain variables. The connected edge-midpoint surface is taken as the physical midsurface: the first fundamental form is evaluated on planar B faces, while a W-centered second fundamental form is constructed from variations of neighboring B-face normals, so membrane and bending share the same geometric carrier. A variational-kernel analysis shows that smooth second-order consistency does not eliminate lattice-scale blind modes. After quotienting out the raw checkerboard gauge, the B metric has one physical membrane blind direction and the symmetric W curvature has two curvature blind directions. We introduce a quotient-minimal membrane compatibility coordinate $X_M$ and an objective reference-relative curvature coordinate $X_W^{rel}$, placed consistently in the $O(t)$ membrane and $O(t^3)$ bending sectors. The formulation admits an explicit midpoint quotient, complete flat blind-mode classification, rigid-motion objectivity, reference-state consistency, and an $O(h^2)$ near-isometry approximation result for aligned generalized cylinders. Numerical tests show second-order curvature convergence, targeted removal of the membrane defect, and a sub-percent, refinement-decaying influence of $X_W^{rel}$. Linear and nonlinear shell benchmarks further demonstrate flat bending, curved-shell membrane-bending coupling, thickness sensitivity, large rotation, nonlinear pinching, and localized ovalization within a single position-only framework.

math.NA↗

arXiv2026

Mesh-Uniform Power Stability of Two-Relaxation-Time Vector Lattice Boltzmann Schemes with Reversible Boundaries

We consider the collision-transport operator for vector-valued two-relaxation-time lattice Boltzmann schemes linearized about a uniform rest state. Assume that the equilibrium blocks are positive definite and that the link-even and link-odd relaxation parameters satisfy s_+ + s_- = 2 and 0 < s_- < 2. If the homogeneous transport is unitary in the equilibrium metric and reversible under velocity exchange, then the powers of the amplification operator are bounded uniformly with respect to the number and arrangement of lattice nodes. The admissible transports include periodic transport, vector halfway bounce-back, coordinate-aligned specular reflection, tangential orthogonal involutions, and compatible multi-channel scattering. The proof reduces the population equation to a two-step macroscopic recurrence generated by a contraction. An inclusion of the numerical range in an ellipse, combined with the Crouzeix-Palencia theorem, yields a dimension-independent estimate for the companion operator and hence the population bound. For a three-coefficient off-midpoint boundary interpolation, we give an exact rational D2N5 example whose finite-domain amplification matrix has a real eigenvalue larger than one, although the interpolation coefficients and bulk parameters are admissible. Thus coefficient convexity alone does not ensure stability for this boundary family.

math.NA↗

arXiv2026

Parameter-Robust Subspace Correction with Multiple Semidefinite Penalties

Independently weighted semidefinite penalties arise in augmented-Lagrangian and constrained formulations. This paper characterizes when an exact additive subspace-correction preconditioner remains uniformly effective over all nonnegative penalty weights on a fixed finite-dimensional space. Robustness holds precisely when the correction spaces decompose every joint kernel generated by a nonempty subset of penalties. If one condition fails, a computable constant determines the exact first-order decay of the smallest preconditioned eigenvalue along the associated parameter ray, and the condition number grows linearly; none of the subset conditions can be discarded in general. Filtered decompositions provide computable sufficient bounds on parameter-ordering cones, while distributive kernel lattices permit a single common splitting. Exact-additive computations confirm the characterization and predicted rates. Separate Scott-Vogelius experiments produce stable multilevel iteration counts over the tested weights and mesh levels. The analysis does not establish mesh-uniformity.

math.NA↗

arXiv2026

The specular ellipse method for scalar ordinary differential equations: exactness and accuracy up to fourth order

This paper introduces a family of one-step implicit methods for solving scalar ordinary differential equations. At each step, the update uses a scaled angular mean of two vector-field evaluations, and the positive scale may vary from step to step. The leading terms of the local truncation error can be expressed in terms of the derivative of the signed curvature of the scaled solution graph. We prove that the proposed method reproduces the exact solution at the mesh points when the solution graph has constant signed curvature under a fixed positive scaling and each implicit update is unique. When this special geometric condition is not satisfied, we establish second-order consistency and convergence for positive scale sequences satisfying suitable uniform conditions. Furthermore, third- and fourth-order consistency and convergence can be achieved by choosing the scale to cancel the relevant curvature terms in the local truncation error. Using only the given problem data, we classify when these improvements are possible and determine the corresponding scale choices. An example shows that the proposed fourth-order method can yield smaller errors than the classical fourth-order Runge--Kutta method at the same step size.

math.NA↗

arXiv2026

From multi-layered problems to multiple two-layered problems: a novel frequency-time hybrid multiple-scattering integral equation solver

This paper proposes a novel frequency-time hybrid multiple scattering (FTH-MS) integral equation solver for time-dependent wave equation problems in general multi-layered media, with remarkable scalability with respect to the number of layers $N$. In light of the finite speed of wave propagation, the new methodology provides an innovative multiple-scattering idea of re-modeling the original $N$-layered problem into a sequence of $N-1$ two-layered sub-problems, for which the main advantages lie in that (i) each sub-problem enjoys much simpler wave scattering properties compared with the complicated problem in a multi-layered medium, (ii) it enables to develop high-accuracy solver utilizing Fourier transform and frequency-domain boundary integral equation (BIE) method; and (iii) numerical evaluation of the sub-problems in each multiple scattering step can be parallelized. Both multiplicative- and additive-type strategies are developed and equivalence results, which indicate that the $M$-th order multiple scattering sums can provide equivalent representations of the solutions up to a certain time $T(M)$, are rigorously derived. Owing to the existed result of exponential convergence of the perfectly-matched-layer (PML) truncation for two-layered problem, all the sub-problems is numerically resolved by means of the FTH method based on the Fourier transform and the PML-BIE method whose numerical evaluation is addressed utilizing the Chebyshev-based rectangular-polar solver with high accuracy. Numerical examples are presented to validate the efficiency and accuracy of the proposed method.

math.NA↗

arXiv2026

A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem

We introduce and analyze a barrier-regularized symmetric Nitsche method for the scalar Signorini problem. Applying a logarithmic barrier to the nonnegative slack variable in an augmented Lagrangian and then eliminating that variable yields a smooth positive-part operator and the perturbed complementarity relation on a primal-dual central path, with barrier parameter $μ=γs$. For every $s>0$, the discrete problem is continuously differentiable, uniquely solvable, and has a symmetric positive definite Newton matrix when the Nitsche parameter is sufficiently large. Under a mild barrier-feasibility condition, the continuous logarithmic energy has a unique minimizer $u_μ$ for every $μ>0$, its gap is positive almost everywhere, and $\|u-u_μ\|_{H^1(Ω)}\lesssimμ^{1/2}$. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the $L^2$ central-path boundary law. If the obstacle is locally the trace of an $H^2$-function, we also prove a uniform local $H^2$ bound in the interior of each planar contact face, on neighborhoods that may contain a free-boundary point of the limiting Signorini solution. The method is exactly consistent and quasi-optimal relative to $u_μ$, with constants independent of $s$. If $u_μ$ is uniformly bounded in $H^r(Ω)$, $3/2<r\leq k+1$, then $\|u-u_h\|_{H^1(Ω)}\lesssim h^{r-1}\|u_μ\|_{H^r(Ω)}+μ^{1/2}$; hence the sufficient balance $s\lesssim h^{2r-1}$ preserves the available energy-norm rate. We also derive rates for discrete penetration and the complementarity residual. Numerical experiments with $P_1$ and $P_2$ elements on structured and unstructured meshes support the predicted rates and the local Newton theory, while showing that a rate-preserving smoothing may still resolve the contact set poorly.

math.NA↗

arXiv2026

Magnetic Field Conforming Multiscale Formulations for Locally-Confined Nonlinear Eddy Current Problems Using the FE-HMM Method

Magnetic composites used for the conversion of electrical energy often incorporate ferromagnetic inclusions insulated from each other to mitigate eddy current losses. Numerical models for these composites must be robust enough to address potential convergence issues arising from the presence of nonlinear magnetic inclusions and presence of significant confined eddy currents in the cell. This paper introduces an $\mathbf{h}$-conforming multiscale formulation for magnetic composites in a periodic framework. The proposed method uses the Heterogeneous Multiscale Method (HMM) with two mesoscale problems: a magnetoquasistatic problem for upscaling the homogenized magnetic flux density $\mathbf{B}_M$ and a magnetostatic problem for upscaling the macroscale incremental reluctivity $(\partial \mathbf{B}_M/\partial \mathbf{H}_M)$. Additionally, the method uses relaxed Newton--Raphson schemes at both macro and mesoscale levels to mitigate the well-known NR convergence issue linked to nonlinear BH constitutive laws in $\mathbf{h}$-conforming formulations. The accuracy and performance of the formulation are evaluated using 2D and 3D idealized periodic soft magnetic composites with linear and nonlinear BH curves. Furthermore, the paper demonstrates that the magnetoquasistatic mesoscale problem can be replaced by a magnet a

math.NA↗

arXiv2026

Efficient primal--dual splitting methods for a Poisson-constrained JKO scheme for Poisson-Nernst-Planck models

The Poisson--Nernst--Planck (PNP) equations strongly couple ionic transport and electrostatic interactions through the Poisson equation, posing substantial numerical challenges under small permittivity and complex potential boundary conditions. Underlying these equations is a natural Wasserstein gradient-flow structure, in which the Poisson equation serves as a local realization of the nonlocal electrostatic interaction energy. Exploiting this structure, we formulate each time step as a constrained convex minimization problem where the ionic continuity equations and the Poisson equation are incorporated as linear constraints, allowing the concentrations, fluxes, and electrostatic potential to be updated simultaneously. The variational structure of the scheme intrinsically guarantees the dissipation of the original free energy, mass conservation, and nonnegativity of ionic concentrations under general electrostatic boundary conditions. Moreover, the framework is structurally modular: extending from classical to modified PNP models with steric interactions and concentration-gradient corrections requires only modifying the energy functional, while all structure-preserving properties are automatically retained. To efficiently solve the resulting large-scale constrained problems, we develop preconditioned and transformed primal--dual algorithms equipped with tailored fast dual solvers, namely DCT-based direct and Schur-complement iterative methods, that exploit the coupled block structure of the PDE constraints. Numerical experiments on classical and modified PNP systems demonstrate the accuracy and structure-preserving properties of the scheme, and show that the proposed algorithms converge reliably in strongly coupled small-permittivity regimes without significant growth in computational cost.

math.NA↗

arXiv2026

Integral Invariants of Vasudeva Murthy's Relaxation Systems: Analysis and Numerical Validation

Vasudeva Murthy's relaxation approach [A.S. Vasudeva Murthy, J. Comput. Appl. Math., 203(2), pp. 437-443, 2007], originally proposed for the Jin-Xin relaxation model, provides an alternative formulation with invariant properties that is consistent and retains the semilinear structure incomparison to the standard one. In this work, Vasudeva Murthy's relaxation approach for various relaxation systems are proposed such as the shallow water equations, the Broadwell model, the Euler equations with heat transfer and two-dimensional Jin-Xin model. For proposed relaxation models, the associated integral invariants are rigorously established at the theoretical level. The main advantage of the integral invariant is that it provides a conserved quantity for the relaxation system by incorporating the coupled contributions of the solution variables in vector form. To validate the analytical results, numerical simulations are carried out for each model using three second-order numerical schemes: CS-EBT2, a semi-implicit second-order central finite-volume scheme for hyperbolic systems with relaxation source terms [S. Sahu, E. Macca, and R. Samala, J. Comput. Phys., 563, 115100, 2026]; UCS2, a finite-volume central relaxation-type scheme [S. F. Liotta, V. Romano, and G. Russo, SIAM J. Numer. Anal., 38(4), 1337-1356, 2000]; and IMEX-RK2, a second-order Implicit-Explicit Runge-Kutta scheme [Pareschi and Russo, J. Sci. Comput., 25, 129-155, 2005]. Numerical results, compared to exact or finely resolved reference solutions, confirm that the models preserve integral invariants, remain stable under CFL restrictions, and exhibit robust and accurate behavior across all benchmark systems tested.

math.NA↗