We study the finite-size Kuramoto model of all-to-all coupled phase oscillators with heterogeneous natural frequencies and characterize the minimal coupling strength required for the existence of a fully phase-locked equilibrium (in a co-rotating frame). To remove the degeneracy due to uniform phase shifts, we move to a reduced co-rotating frame and assess stability through the Jacobian of the reduced system: a fully phase-locked state is stable when this Jacobian is negative definite. This defines a stability region in the phase space. The Kuramoto vector field maps this region to a convex set in frequency space, so a fully-locked state at coupling K exists exactly when the rescaled frequency vector /K lies inside that convex image. The critical coupling K_ℓ is defined as the smallest coupling strength for which a fully phase-locked equilibrium exists; geometrically, it corresponds to the first intersection of the ray t with the boundary of this convex set. Building on this convex-geometric structure, we construct an explicit polytope from analytically computable boundary points of the stability region, providing a closed-form upper bound K_b ≥ K_ℓ. The bound is exact for frequencies aligned with polytope vertices and offers a fully explicit outer approximation for general frequency vectors. While not uniformly sharp in a quantitative sense, this construction exposes the underlying geometry of stable fully phase-locking solutions. These results provide a practical use the convex-geometric structure underlying stable fully-locked states in the Kuramoto model.

This work extends the discrete compactness results of Walkington (SIAM J. Numer. Anal., 47(6):4680-4710, 2010) for high-order discontinuous Galerkin time discretizations of parabolic problems to more general function space settings. In particular, we show a discrete version of the Aubin-Lions-Simon lemma that holds for general Banach spaces X, B, and Y satisfying X hooked right arrow B compactly and B hooked right arrow Y continuously. Our proofs rely on the properties of a time reconstruction operator and remove the need for quasi-uniform time partitions assumed in previous works. Thus, we provide a useful and flexible tool for the analysis of high-order discontinuous Galerkin time discretizations of complex nonlinear partial differential equations.

We present and analyze an embedded Trefftz discontinuous Galerkin method for reaction-diffusion problems on anisotropic meshes. The method is constructed by imposing a relaxed local Trefftz condition via an embedding into a tensor-product DG space, yielding a reduced global system while preserving the approximation properties of the underlying high-order discretization. We prove stability and quasi-optimality on anisotropic, possibly curved, quadrilateral elements, and derive anisotropic a priori error estimates. Numerical experiments for h- and hp-refinement, including curved-domain examples, validate the theoretical results.

We propose and analyze a class of finite element methods for the time-dependent incompressible magnetohydrodynamics system based on H(curl)-conforming discretizations for both the velocity and the magnetic field. This choice is guided by the aim of developing methods that are also suitable for the types of solutions arising in problems posed on nonconvex domains. Within this framework, we introduce three stabilized formulations, and study how the stabilization mechanisms employed influence their structural properties. In particular, we focus on suitability for nonconvex polyhedral domains, the need for Lagrange multipliers for the magnetic field, pressure-robustness, and quasi-robustness with respect to both the fluid and magnetic Reynolds numbers. The proposed formulations are further assessed through numerical experiments, highlighting their practical performance.

We formulate, analyse, and implement a discontinuous Galerkin finite element method (DG-FEM) for the approximation of the solution of an elliptic boundary value problem in a domain with fractal boundary. We consider the case of the Poisson equation in the Koch snowflake domain with zero Dirichlet boundary conditions, but our methodology can be generalised to other cases. Rather than first approximating the snowflake domain by a polygonal "prefractal” and then applying a standard DG-FEM on the prefractal, we define a DG-FEM on the snowflake itself, using a geometry-conforming mesh (a fractal tiling) consisting of fractal elements, each similar to the original snowflake. Fluxes across inter-element boundaries, which are fractal curves, are represented in a weak way by integrals over element subdomains. We show how, for local polynomial basis functions, these integrals can be evaluated exactly using the similarity of the elements. We prove well-posedness and quasi-optimality of the method, and provide a partial convergence analysis. We present numerical results for piecewise linear and piecewise quadratic basis functions, which demonstrate the effectiveness of the method.

The Compact Discontinuous Galerkin method was introduced by Peraire and Persson in (SIAM J. Sci. Comput., 30, 1806–1824, 2008). In this work, we present the stability and convergence analysis for the hp-version of this method applied to elliptic problems on polytopal meshes. Moreover, we introduce fast and practical algorithms that allow the CDG, LDG, and BR2 methods to be implemented within a unified framework. Our numerical experiments show that the CDG method yields a compact stencil for the stiffness matrix, with faster assembly and solving times compared to the LDG and BR2 methods. We numerically study how coercivity depends on the method parameters for various mesh types, with particular focus on the number of facets per mesh element. Finally, we demonstrate the importance of choosing the correct directions for the numerical fluxes when using variable polynomial degrees.

We present a unified framework for the analysis of space-time methods based on Galerkin-type time discretizations for parabolic and hyperbolic problems. Crucially, the stability analysis relies on a suitable choice of test functions to establish the continuous dependence of the discrete solution on the data in L^∞(0, T; X) norms, which is then used to derive a priori error estimates. This approach closes the gap in the analysis of some methods in this class caused by the limitation of standard energy arguments, and is characterized by the absence of Grönwall estimates, applicability to arbitrary approximation degrees, reduced regularity assumptions, and robustness with respect to the model parameters.

We present and analyse a structure-preserving method for the approximation of solutions to nonlinear cross-diffusion systems, which combines a Local Discontinuous Galerkin spatial discretization with the backward Euler time-stepping scheme. The proposed method makes use of the underlying entropy structure of the system, expressing the main unknown in terms of the entropy variable by means of a nonlinear transformation. Such a transformation allows for imposing the physical positivity or boundedness constraints on the approximate solution in a strong sense. A key advantage of our scheme is that nonlinearities do not appear explicitly within differential operators or interface terms in the scheme, which significantly improves its efficiency and eases its implementation. We prove the existence of discrete solutions and their asymptotic convergence to a weak solution to the continuous problem. Numerical results for some one- and two-dimensional problems illustrate the accuracy and entropy stability of the proposed method.

Higher-order interactions have typically been modeled using hypergraphs or simplicial complexes, where interactions explicitly involve more than two nodes. Here we demonstrate that effective higher-order dynamical constraints emerge naturally on ordinary graphs, provided the interaction carries nontrivial topological structure. We study a gauge-coupled phase model with edge phase lags whose accumulation around closed loops produces gauge-invariant mismatches. We show that the associated twisted Laplacian admits a zero mode if and only if all cycle holonomies vanish. Consequently, global compatibility is obstructed not by local pairwise mismatches, but by intrinsic topological frustration on cycles. We then connect this framework to the symmetric Sakaguchi–Kuramoto model, whose local coupling law differs from the gauge-coupled model but whose node phases impose cycle closure on physical phase differences. For cactus graphs, path mismatches induced by the symmetric lag can be represented through associated cycle holonomies, providing a static spectral encoding of their global residual incompatibility. Our results establish a spectral framework linking frustration to cycle-level constraints and identify cycle holonomy as a local-to-global diagnostic of path incompatibility in synchronization dynamics.

This work presents a structure-preserving, high-order, unconditionally stable numerical method for approximating the solution to the Fisher-Kolmogorov equation on polytopal meshes, with a particular focus on its application in simulating misfolded protein spreading in neurodegenerative diseases. The model problem is reformulated using an entropy variable to guarantee solution positivity, boundedness, and satisfaction of a discrete entropy-stability inequality at the numerical level. The scheme combines a local discontinuous Galerkin method on polytopal meshes for the space discretization with a nu-step backward differentiation formula for the time integration. Implementation details are discussed, including a detailed derivation of the linear systems arising from Newton's iteration. The accuracy and robustness of the proposed method are demonstrated through extensive numerical tests. Finally, the method's practical performance is demonstrated through simulations of nu-synuclein propagation in a two-dimensional brain geometry segmented from MRI data, providing a relevant computational framework for modeling synucleinopathies (such as Parkinson's disease) and, more generally, neurodegenerative diseases.

We propose and analyze a space–time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space–time meshes. Existence and uniqueness of a discrete solution are shown by means of an inf-sup condition, whose proof does not rely on polynomial inverse estimates. Moreover, for piecewise polynomial spaces satisfying an additional mild condition, we show a second inf-sup condition that provides additional control over the time derivative of the discrete solution. We derive hp-a priori error bounds based on these inf-sup conditions, which we use to prove convergence rates for standard, tensor-product, and quasi-Trefftz polynomial spaces. Numerical experiments validate our theoretical results.

We propose and analyze a space-time finite element method for Westervelt's quasilinear model of ultrasound waves in its second-order formulation. The method combines conforming finite element spatial discretizations with a discontinuous-continuous Galerkin time stepping. Its analysis is challenged by the fact that standard Galerkin testing approaches for wave problems do not allow for bounding the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach's fixed-point argument, we show the well-posedness of the scheme, a priori error estimates, and robustness with respect to the strong damping parameter delta. Moreover, the scheme preserves the asymptotic preserving property of the continuous problem; more precisely, we prove that the discrete solutions corresponding to delta > 0 converge, in the singular vanishing dissipation limit, to the solution of the discrete inviscid problem. We use several numerical experiments in (2+ 1) dimensions to validate our theoretical results.

We carry out a stability and convergence analysis for the fully discrete scheme obtained by combining a finite or virtual element spatial discretization with the upwind-discontinuous Galerkin time-stepping applied to the time-dependent advection-diffusion equation. A space- time streamline-upwind Petrov-Galerkin term is used to stabilize the method. More precisely, we show that the method is inf-sup stable with constant independent of the diffusion coefficient, which ensures the robustness of the method in the convection- and diffusion-dominated regimes. Moreover, we prove optimal convergence rates in both regimes for the error in the energy norm. An important feature of the presented analysis is the control in the full L2(0, T; L2(Q)) norm without the need of introducing an artificial reaction term in the model. We finally present some numerical experiments in (3 + 1)-dimensions that validate our theoretical results.

We discuss the asymptotic-preserving properties of a hybridizable discontinuous Galerkin method for the Westervelt model of ultrasound waves. More precisely, we show that the proposed method is robust with respect to small values of the sound diffusivity damping parameter delta by deriving low- and high-order energy stability estimates, and a priori error bounds that are independent of delta. Such bounds are then used to show that, when delta -> 0+, the method remains stable and the discrete acoustic velocity potential psi h(delta)psi h(delta)$ \psi_h<^>{(\delta)}$ converges to psi h(0)psi h(0)$ \psi_h<^>{(0)}$, where the latter is the singular vanishing dissipation limit. Moreover, we prove optimal convergence rates for the approximation of the acoustic particle velocity variable upsilon=del psi upsilon-=del psi$ \underset-{\boldsymbol\upsilon}=\nabla\psi$. The established theoretical results are illustrated with some numerical experiments.

This work extends the discrete compactness results of Walkington (SIAM J. Numer. Anal., 47(6):4680–4710, 2010) for high-order discontinuous Galerkin time discretizations of parabolic problems to more general function space settings. In particular, we show a discrete version of the Aubin–Lions–Simon lemma that holds for general Banach spaces X, B, and Y satisfying X ↪ B compactly and B ↪ Y continuously. Our proofs rely on the properties of a time reconstruction operator and remove the need for quasi-uniform time partitions assumed in previous works. Thus, we provide a useful and flexible tool for the analysis of high-order discontinuous Galerkin time discretizations of complex nonlinear partial differential equations.

"mdendro" is an R package that provides a comprehensive collection of linkage methods for agglomerative hierarchical clustering on a matrix of proximity data (distances or similarities), returning a multifurcated dendrogram or multidendrogram. Multidendrograms can group more than two clusters at the same time, solving the nonuniqueness problem that arises when there are ties in the data. This problem causes that different binary dendrograms are possible depending both on the order of the input data and on the criterion used to break ties. Weighted and unweighted versions of the most common linkage methods are included in the package, which also implements two parametric linkage methods. In addition, package "mdendro" provides five descriptive measures to analyze the resulting dendrograms: cophenetic correlation coefficient, space distortion ratio, agglomerative coefficient, chaining coefficient and tree balance.

We present a stability and convergence analysis of the space-time continuous finite element method for the Hamiltonian formulation of the wave equation. More precisely, we prove a continuous dependence of the discrete solution on the data in a C0([0, T]; X)-type energy norm, which does not require any restriction on the meshsize or the time steps. Such a stability result is then used to derive a priori error estimates with quasi-optimal convergence rates, where a suitable treatment of possible nonhomogeneous Dirichlet boundary conditions is pivotal to avoid loss of accuracy. Moreover, based on the properties of a postprocessed approximation, we derive a constant-free, reliable a posteriori error estimate in the C0([0, T]; L2(Omega)) norm for the semidiscrete-intime formulation. Several numerical experiments are presented to validate our theoretical findings.

We investigate a two-state conformational conversion system and introduce a novel structure-preserving numerical scheme that couples a local discontinuous Galerkin space discretization with the backward Euler time-integration method. The model is first reformulated in terms of auxiliary variables involving suitable nonlinear transformations, which allow us to enforce positivity and boundedness at the numerical level. Then, we prove a discrete entropy-stability inequality, which we use to show the existence of discrete solutions, as well as to establish the convergence of the scheme by means of some discrete compactness arguments. As a by-product of the theoretical analysis, we also prove the existence of global weak solutions satisfying the system's physical bounds. Numerical results validate the theoretical results and assess the capabilities of the proposed method in practice.

Models of network diffusion typically rely on the Laplacian matrix, capturing interactions via direct connections. Beyond direct interactions, information in many systems can also flow via indirect pathways, where influence typically diminishes over distance. In this work, we analyze diffusion dynamics incorporating such indirect connections using the d-path Laplacian framework. We introduce a parameter, the indirect influence, based on the change in the second smallest eigenvalue of the generalized path Laplacian, to quantify the impact of these pathways on diffusion timescales relative to direct-only models. Using perturbation theory and mean-field approximations, we derive analytical expressions for the indirect influence in terms of structural properties of random networks. Theoretical predictions align well with numerical simulations, providing a phase diagram for when indirect influence becomes significant. We also identify a structural phase transition governed by the emergence of d-paths and derive the critical connection probability above which they dramatically alter diffusion. This study provides a quantitative understanding of how indirect pathways shape network dynamics and reveals their collective structural onset.

We propose a space-time isogeometric finite element method for the linear Schrödinger equation, and establish its unconditional stability through a matrix-based analysis. Although maximal-regularity splines in time provide higher accuracy per degree of freedom compared to piecewise continuous polynomials, the nonlocal support of the spline bases precludes the use of standard variational arguments in the stability proofs. To overcome this, we show that the resulting scheme is governed by a family of nearly Toeplitz system matrices and, by studying the condition number of these matrices, we prove that the family is weakly well-conditioned, which guarantees the unconditional stability of the method. Furthermore, the discrete scheme preserves mass and energy at the final time. Numerical experiments confirm our theoretical findings and illustrate the optimal convergence behavior of the scheme. Finally, we exploit an algebraic connection between our formulation and a recent first-order-in-time space-time isogeometric method for the wave equation to derive a complete matrix-based stability analysis for the latter.