Elena Bernardelli, Elena Gaburro, Michael Dumbser
We present a novel structure-preserving semi-implicit finite volume method on vertex-based staggered meshes for the compatible discretization of first order systems of time-dependent partial differential equations (PDEs). The method preserves divergence-free and curl-free vector fields exactly thanks to the compatible vertex-staggered discretization of the state variables on unstructured grids that are constituted by primal Delaunay triangles and their dual polygons. For the weakly compressible Euler equations, the scheme is asymptotic preserving, yielding a consistent discretization of the incompressible limit as the Mach number goes to zero. The new scheme applies to a broad spectrum of PDEs, including the weakly compressible and incompressible Euler and Navier-Stokes equations, the incompressible magnetohydrodynamics (MHD) system, and the incompressible version of the first-order hyperbolic Godunov-Peshkov-Romenski (GPR) model for continuum mechanics. The computational domain is covered by a primal triangular mesh and a dual tessellation made of so-called star polygons. Scalar quantities (pressure, density, viscous stress) are defined at nodes, with pressure updated implicitly in a continuous finite element fashion, yielding a symmetric and positive definite pressure system. Instead, vector fields (velocity, momentum, magnetic and distortion fields) are stored at triangle barycenters and evolved explicitly using a compatible finite volume scheme. Thanks to the semi-implicit discretization, the CFL condition is independent of the sound speed, allowing simulations at low Mach numbers. The fully compatible formulation ensures exactly divergence-free velocity field in the incompressible limit, exactly divergence-free magnetic field for MHD, and exactly curl-free inverse deformation gradient in solid mechanics. The method is validated through a wide set of test cases.
Benedikt Gräßle
A Newton--Kantorovich-type argument enables the a posteriori existence verification of a unique regular root near a computed approximation, purely from computable data. This framework allows for non-selfadjoint problems and extends the existing verification theory to nonconforming discretisations. A key ingredient is a guaranteed lower bound on the continuous inf-sup constant from a quasi-optimal nonconforming discretisation that enables a novel a priori error estimator. All quantities are obtained by post-processing a single discretisation; convergence rates are proved. The theory is applied to a fourth-order formulation of the stationary two-dimensional Navier--Stokes equations and illustrated by numerical experiments.
Miha Rot, Gregor Kosec
One of the main challenges in numerically solving partial differential equations is finding a discretisation for the computational domain that balances the accurate representation of the underlying field with computational efficiency. Meshless methods approximate differential operators based on the values of the field in computational nodes, offering a natural approach to adaptivity. The density of computational nodes can either be increased to enhance accuracy or decreased to reduce the number of numerical operations, depending on the properties of the intermediate solution. In this paper, we utilise an adaptive discretisation approach for the numerical simulation of natural convection in non-Newtonian fluid flow. The shear-thinning behaviour is interesting both due to its numerous occurrences in nature, blood being a prime example, and due to its properties, as the decreasing viscosity with increasing shear rate results in sharper flow structures. We focus on the de Vahl Davis test case, a natural convection driven flow in a differentially heated rectangular cavity. The thin boundary layer flow along the vertical boundaries makes this an ideal test case for refinement. We demonstrate that adaptively refining the node density enhances computational efficiency and examine how the parameters for adaptive refinement affect the solution.
Di Wu, Ling Liang, Haizhao Yang
Bayesian Optimal Experimental Design (BOED) provides a rigorous framework for decision-making tasks in which data acquisition is often the critical bottleneck, especially in resource-constrained settings. Traditionally, BOED typically selects designs by maximizing expected information gain (EIG), commonly defined through the Kullback-Leibler (KL) divergence. However, classical evaluation of EIG often involves challenging nested expectations, and even advanced variational methods leave the underlying log-density-ratio objective unchanged. As a result, support mismatch, tail underestimation, and rare-event sensitivity remain intrinsic concerns for KL-based BOED. To address these fundamental bottlenecks, we introduce an IPM-based BOED framework that replaces density-based divergences with integral probability metrics (IPMs), including the Wasserstein distance, Maximum Mean Discrepancy, and Energy Distance, resulting in a highly flexible plug-and-play BOED framework. We establish theoretical guarantees showing that IPM-based utilities provide stronger geometry-aware stability under surrogate-model error and prior misspecification than classical EIG-based utilities. We also validate the proposed framework empirically, demonstrating that IPM-based designs yield highly concentrated credible sets. Furthermore, by extending the same sample-based BOED template in a plug-and-play manner to geometry-aware discrepancies beyond the IPM class, illustrated by a neural optimal transport estimator, we achieve accurate optimal designs in high-dimensional settings where conventional nested Monte Carlo estimators and advanced variational methods fail.
Zahra Monfared, Saksham Malhotra, Sekiya Hajime, Ioannis Kevrekidis, Felix Dietrich
For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic numerical computation of the eigenspaces of the operator. Given a small set of (so-called ``principal'') eigenfunctions that are approximated conventionally, we can obtain a much larger set by constructing polynomials of the principal eigenfunctions. This enriches the set, and thus allows us to more accurately represent application-specific observables. Often, eigenfunctions exhibit localized singularities (e.g. in simple, one-dimensional problems with multiple steady states) or extended ones (e.g. in simple, two-dimensional problems possessing a limit cycle, or a separatrix); we discuss eigenfunction matching/continuation across such singularities. By handling eigenfunction singularities and enabling their continuation, our approach supports learning consistent global representations from locally sampled data. This is particularly relevant for multistable systems and applications with sparse or fragmented measurements.
Sophie Moufawad, Nabil Nassif, Faouzi Triki
We consider the mathematical model of gas trapping in deep polar ice (firns), which consists of a parabolic partial differential equation, that can degenerate at one boundary extreme. In [1], we considered all the coefficients to be constants, except the diffusion coefficient D(z) that is to be reconstructed. In this paper, we assume both the diffusion coefficient D(z) and the volume fraction f(z) are functions. The difficulty in this problem, both theoretically and computationally, arises from the fact that D(z) and f(z) may be zero at bottom of the firn. To handle such degeneracy, we defined appropriate weighted Sobolev spaces and used Lion's theorem to prove existence and uniqueness of the semi-variational formulation of the Firn PDE. A full discrete system is obtained through a P1 Finite element Galerkin procedure in space and an Euler-Implicit scheme in time. Sufficient conditions for the existence and uniqueness of the solution for the discrete system are obtained.
Yue Zhao, Daochang Zhang, Dijana Mosic
This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal A-\mathcal C*_N\mathcal D^D*_N\mathcal B$ via the Einstein product, formulated using the Drazin inverse of $\mathcal A$ and the generalized Schur complement $\mathcal D-\mathcal B*_N\mathcal A^{D}*_N\mathcal C$, providing a comprehensive generalization and unification of existing results in the literature for the case when the tensors are of order two. Furthermore, the findings reduce to the classical Sherman-Morrison-Woodbury formula in the special case of second-order tensors. Finally, we give an example to illustrate our new explicit expression.
Reinhard Nabben, Ludwig Rooch
Recently a new approach to analyze and create algebraic multigrid methods (AMG) for nonsymmetric and indefinite matrices was established. Convergence is measured in general norms induced by a certain HPD matrix $B$ and $B$-orthogonal projections built by compatible transfer operators are used. Here we continue our theoretical framework, started in Nabben and Rooch (2026), for nonsymmetric algebraic multigrid methods using any HPD matrix $B$ to induce a norm. Our framework not only includes all recent results but also provides many new results. We consider two, slightly different, multigrid operators. The first one is the natural generalization of the error operator in the HPD case. The second operator is simpler to apply and has been studied before. However, an additional condition for the smoother $M^{-1}A$ is needed, which is in our terminology the $B$-normality. We explain the differences and similarities of both operators in detail and show, why the extra condition is needed. We consider arbitrary interpolation and restriction operators that result in $B$-orthogonal coarse-grid corrections and give sharp estimates for the norm of the error propagation matrices for the two-grid methods. We also show, that the norms are decreasing if we increase the size of the coarse space. Moreover, we are able to extend the landmark $V$-cycle bound by McCormick to the nonsymmetric case.
Qinchen Song, Lei Zhang, Min Tang
When solving the time-dependent radiative transport equation (RTE), implicit time discretization is often employed for its robustness and stability. This results in a sequence of steady-state RTEs with identical cross-sections but varying source terms, whose repeated solution is computationally costly. To address this, we first apply the adaptive tailored finite point scheme (TFPS) for spatial discretization. This scheme exploits prior knowledge of the background media's optical properties to adaptively compress the angular domain, constructing a compressed linear system. A key feature is its ability to reconstruct the layer structure after compression, faithfully capturing the variance at the layer. We then use the Recursive Skeleton Method (RSM) to obtain an explicit multilevel decomposition of the inverse discrete operator, which is reused for all steady-state solutions. Numerical experiments show that our framework achieves high accuracy and significant efficiency across diverse scenarios.
Erik Burman, Fabian Heimann
In this paper, we investigate the combination of a linear continuous interior penalty type and a non-linear artificial diffusion stabilisation applied to the transport problem, based on continuous Galerkin finite elements in space. This method was recently introduced and analysed for globally smooth solutions in [Burman 2023, SIAM J. Sci. Comput., 45, 1, A96-A122]. We provide a rigorous proof of a localisation principle in terms of weighted stability and a priori error bound results, which follow the widely known $\mathcal{O}(h^{k+1/2})$ scaling in the $L^2(Ω; t=T)$ norm, where $k$ denotes the polynomial order of the finite element space and $h$ the mesh size. The analysis is semi-discrete in space and assumes sufficient local regularity of the continuous solution on the smooth part of the domain, where the continuous interior penalty stabilisation is active, whilst artificial diffusion operates on the remaining rough parts of the domain. Thereby, the analysis demonstrates that typical numerical errors in the rough part stay localised relative to the convection velocity and do not negatively affect the smooth parts of the solution, if the stabilisation combination is set up accordingly.
Anita Gjesteland, Sigrun Ortleb, Salim Elghawi, David C. Del Rey Fernández
We develop an unconditionally energy-stable tensor-product space-time discretization framework for the solution of a linear kinetic transport equation in one space dimension. The kinetic equation is a simplified model of radiative transfer formulated as a hyperbolic balance law in diffusive scaling for a particle distribution function of the independent variables space, time and velocity. Our numerical discretization is based on the well-known technique of micro-macro decomposition which results in a system of balance laws for equilibrium and non-equilibrium quantities and facilitates preservation of the asymptotic limit for vanishing scaling parameters at the discrete level. We prove fully discrete stability and asymptotic preservation for general spatial and temporal discretizations having the summation-by-parts property. A new provably energy-stable Dirichlet boundary treatment for the micro-macro decomposed system is developed based on the introduction of simultaneous approximation terms. Numerical results show convergence for smooth problems and demonstrate energy stability of the proposed boundary treatment.
Gang Chen, Daozhi Han, Jiaxuan Liu, Yangwen Zhang, Dujin Zuo
We propose a hybridizable discontinuous Galerkin (HDG) method combined with convex-concave splitting for the temporal discretization of the convective Cahn-Hilliard equation. The convection term is discretized explicitly without stabilization, yielding three key advantages: (1) unconditional stability, (2) preservation of the optimal convergence rate for piecewise constant approximations, and (3) a symmetric system after local elimination, enabling efficient solver via minimal residual methods. We establish optimal convergence rates in the $L^2$ norm for both the scalar and flux variables for any polynomial degree $k \geq 0$. To achieve optimal $L^2$-norm estimates, we introduce a specialized HDG elliptic projection operator and analyze its approximation properties. Within the HDG framework, local elimination is employed to reduce the degrees of freedom associated with the globally coupled unknowns, and the scalar variables exhibit superconvergence. Finally, numerical experiments validate the theoretical convergence rates and demonstrate the effectiveness of the proposed method.
Shounian Deng, Weiyin Fei, Banban Shi
Most existing literature focuses on pointwise convergence (i.e., convergence at a fixed time point) of numerical solutions for Stochastic functional differential equations (SFDEs). In contrast, this paper investigates the strong segment convergence (i.e., the strong order of convergence of the numerical segment process). For SFDEs with super-linear drift and diffusion coefficients, we employ the explicit truncated Euler-Maruyama (EM) scheme. First, we establish the uniform moment boundedness of the truncated EM solution over a finite time interval. Second, we derive the $L^2$-error estimate between the continuous numerical segment and the step numerical segment. Finally, we prove the strong convergence order of the numerical segment generated by the truncated EM. The results can be used to analyze invariant measures and ergodicity of numerical segment, and have important applications in practical problems such as path-dependent financial options. We also provide a numerical example to support the theoretical results.
Aaron Brunk, Dennis Höhn
This work presents a conforming finite-element scheme for the non-isothermal Allen-Cahn-Navier-Stokes system, incorporating periodic, closed, and thermal boundary conditions. The system comprises the incompressible Navier-Stokes equations coupled with the non-isothermal Allen-Cahn equation, which includes a non-conserved phase-field equation and a temperature equation. The proposed numerical scheme preserves entropy production exactly and maintains total energy conservation up to a negative numerical dissipation. Convergence tests in both space and time are conducted, and representative examples are provided to demonstrate the scheme's effectiveness.
Reinhard Nabben, Ludwig Rooch
Algebraic Multigrid (AMG) methods have been proven to be effective solvers for large-scale linear algebraic systems $Ax = b$ with Hermitian positive definite (HPD) matrix $A$. For such problems the convergence in the $A$-norm is well understood, but for nonsymmetric indefinite systems fewer results exist. Recently, convergence results for more general $B$-norms induced by certain HPD matrices were established. There, orthogonal projections built by compatible transfer operators are used. Here, we present a theoretical framework for the convergence of nonsymmetric algebraic two-grid methods for arbitrary $B$-inner products and induced $B$-norms which naturally includes the HPD case and all recent results for the nonsymmetric case. For this purpose, we consider two different two-grid error operators with the first one being the natural generalization of the error operator in the HPD case. The second operator has been studied before and is simpler, but requires the additional assumption of normality in some inner product of the smoothing step $M^{-1}A$ to achieve convergence. We prove new convergence results, generalize some previous results and explain the differences and similarities of both operators together with the necessity of the normality. Moreover, we establish optimal compatible interpolation and restriction operators for both two-grid methods that minimize the error norm.
Jieling Yang, Guosheng Fu
We extend the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method (ES-OEDG) on curvilinear meshes to adaptive mesh refinement (AMR) grids with nonconforming interfaces. The formulation targets two-dimensional curvilinear quadrilateral meshes under a 2:1 refinement constraint, allowing a single level of hanging nodes. Elementwise volume discretization and geometric mapping are retained, while oscillation elimination and interface coupling are adapted for nonconforming interfaces. A central contribution is the design and analysis of numerical fluxes for such interfaces. We construct an entropy-stable flux that ensures global conservation and a semi-discrete entropy inequality. However, for polynomial degree N >= 2, negative entries in nonconforming interpolation operators lead to loss of formal high-order consistency. To address this, we propose a mortar-based flux that preserves high-order accuracy by interpolating at the solution level and evaluating standard two-point fluxes on fine-side mortars, at the cost of losing provable entropy stability. We also extend the Zhang--Shu positivity-preserving framework to curvilinear AMR meshes. Under forward Euler time stepping and a suitable CFL condition, the scheme using either flux preserves positivity of cell-average density and pressure. Combined with the Zhang--Shu limiter, this yields a fully discrete scheme maintaining admissibility at all nodal points. We further incorporate shock-indicator-based AMR and a conservative, positivity-preserving data transfer procedure between successive meshes, resulting in a robust and efficient algorithm. Numerical experiments on Cartesian and curvilinear AMR grids confirm high-order accuracy and robustness.
Michele Botti, Lorenzo Mascotto, Marialetizia Mosconi
We analyze a dual mixed nonconforming discretization of a generalized Darcy-Forchheimer model. Compared to the analogous scheme proposed by Girault and Wheeler, we consider general, i.e., nonquadratic, Forchheimer nonlinearities; we admit mixed, inhomogeneous boundary conditions; we allow for more general, i.e., with lower Lebesgue regularity, permeability tensors; we construct general-order schemes; we prove convergence to the exact solution under low regularity assumptions, based on novel Sobolev-trace inequalities for broken spaces; we derive error estimates of general-order assuming extra regularity of the exact solution and data; we present numerical results assessing the performance of the proposed schemes for different types of nonlinearity and nonlinear solvers.
Sebastian Celis Sierra, Meruyert Khamitova, Ran Zhao, Sadeed Bin Sayed, Hakan Bagci
A thin-sheet (TS) volume integral equation (VIE) formulation incorporating generalized sheet transition conditions (GSTCs) is presented for the simulation of three-dimensional (3D) bianisotropic metasurfaces. The metasurface is represented as an equivalent TS, with its constitutive tensors derived from the GSTC susceptibility tensors. Invoking the TS approximation, the governing VIEs are reduced to surface integral equations (SIEs), in which tangential and normal flux density components are treated as distinct sets of unknowns and discretized using Rao-Wilton-Glisson and pulse basis functions, respectively. In contrast to conventional GSTC approaches based on conventional SIEs, which represent only tangential fields, the proposed framework rigorously enforces the bianisotropic GSTCs, including normal field interactions, while retaining the flux-based VIE character of the formulation. Numerical examples demonstrate the accuracy and robustness of the proposed TS-VIE-GSTC solver for polarization rotation, perfect reflection, multi-directional attenuation, and oblique phase-shift transformation.
Bodhinanda Chandra, Sachith Dunatunga, Ken Kamrin
This work presents a unified viscoelastic-viscoplastic continuum framework for modeling rate-dependent granular flows across regimes. The formulation incorporates two distinct rate-dependent mechanisms, namely micro-inertia and viscoelastic dissipation, within a single continuum description. A central contribution is an explicit link between the coefficient of restitution and a continuum viscosity, derived from an analysis of wave attenuation in granular assemblies, thereby establishing a direct connection between particle-scale collision physics and macroscopic damping. This relation is introduced while retaining inertia-dependent plastic flow governed by the classical $μ(I)$ rheology. The constitutive model is constructed by meticulously partitioning elastic and viscous responses within the model and corresponding stress-update routine, such that viscous dissipation governs wave propagation and collisional processes without altering the plastic flow rule. The framework is implemented within the material point method to simulate transient processes involving large deformations, material separation, and subsequent reconsolidation. A range of numerical examples, including steady, transient, vibrational, and impact-driven flows, demonstrates that the model captures wave propagation, diffusion, and rate-dependent granular behavior within a unified continuum setting.
Johann S. Brauchart, Josef Dick, Friedrich Pillichshammer
We prove that the information complexity (i.e., the inverse) of the classical spherical cap $L_2$ discrepancy on the $d$-dimensional sphere $\mathbb{S}^d$ decreases with dimension $d$, indicating a ``blessing of dimensionality'' for the associated numerical integration problem. We then introduce a modified spherical cap $L_2$ discrepancy that emphasizes large caps (close to hemispheres). For this variant, the problem does not become easier with increasing $d$. We also establish a Stolarsky invariance principle which connects the modified spherical cap $L_2$ discrepancy to numerical integration in the Sobolev space $H^{(d+1)/2}(\mathbb{S}^d)$, represented by the reproducing kernel $K(\boldsymbol{x}, \boldsymbol{y}) = 1 - \tfrac{1}{\sqrt{2}} \|\boldsymbol{x} - \boldsymbol{y}\|$. Stolarsky's invariance principle then implies that the worst-case integration error in this space grows polynomially with $d$.