Design of Hierarchical Excitable Networks - von der Gracht, Sören; Lohse, Alexander (arXiv:2603.06157)
We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $G_1,\dotsc,G_N$ (the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $G_1,\dotsc,G_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbfΓ$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbfΓ$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $δ$-neighborhood of the first set contains an initial condition with $ω$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.

Homogeneous coupled cell systems with high-dimensional internal dynamics - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (Chaos, Solitons & Fractals)
We investigate homogeneous coupled cell systems with high-dimensional internal dynamics. In many studies on network dynamics, the analysis is restricted to networks with one-dimensional internal dynamics. Here, we show how symmetry explains the relation between dynamical behavior of systems with one-dimensional internal dynamics and with higher dimensional internal dynamics, when the underlying network topology is the same. Fundamental networks of homogeneous coupled cell systems (B. Rink, J. Sanders. Coupled Cell Networks and Their Hidden Symmetries. SIAM J. Math. Anal. 46.2 (2014)) can be expressed in terms of monoid representations, which uniquely decompose into indecomposable subrepresentations. In the high-dimensional internal dynamics case, these subrepresentations are isomorphic to multiple copies of those one computes in the one-dimensional internal dynamics case. This has interesting implications for possible center subspaces in bifurcation analysis. We describe the effect on steady state and Hopf bifurcations in l-parameter families of network vector fields. The main results in that regard are that (1) generic one-parameter steady state bifurcations are qualitatively independent of the dimension of the internal dynamics and that, (2) in order to observe all generic l-parameter bifurcations that may occur for internal dynamics of any dimension, the internal dynamics has to be at least l-dimensional for steady state bifurcations and 2l-dimensional for Hopf bifurcations. Furthermore, we illustrate how additional structure in the network can be exploited to obtain even greater understanding of bifurcation scenarios in the high-dimensional case beyond qualitative statements about the collective dynamics. One-parameter steady state bifurcations in feedforward networks exhibit an unusual amplification in the asymptotic growth rates of individual cells, when these are one-dimensional (S. von der Gracht, E. Nijholt, B. Rink. Amplified steady state bifurcations in feedforward networks. Nonlinearity 35.4 (2022)). As another main result, we prove that (3) the same cells exhibit this amplifying effect with the same growth rates when the internal dynamics is high-dimensional.
Keywords: Coupled cell systems; Network dynamics; Dimension reduction; Bifurcation theory; Symmetry; Monoid representation theory

Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces - Bullerjahn, Nils; Kovács, Balázs (ArXiv)

Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions - Bullerjahn, Nils; Kovács, Balázs (IMA Journal of Numerical Analysis)

Numerical analysis of an evolving bulk--surface model of tumour growth - Edelmann, Dominik; Kovács, Balázs; Lubich, Christian (IMA Journal of Numerical Analysis)

Maximal regularity of evolving FEMs for parabolic equations on an evolving surface - Bai, Genming; Kovács, Balázs; Li, Buyang (IMA Journal of Numerical Analysis)
In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods

Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions - Bullerjahn, Nils (arXiv)

Numerical surgery for mean curvature flow of surfaces - Kovács, Balázs (SIAM Journal on Scientific Computing)

A posteriori error estimates for parabolic partial differential equations on stationary surfaces - Kovács, Balázs; Lantelme, Michael Frederik Raúl (arXiv:2407.02101)
This paper develops and discusses a residual-based a posteriori error estimate and a space--time adaptive algorithm for solving parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. A space--time adaptive algorithm is proposed using the derived error indicator. Numerical experiments illustrate and complement the theory.

Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints - Bartels, Sören; Kovács, Balázs; Wang, Zhangxian (IMA Journal of Numerical Analysis)
Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates - Contri, Alessandro; Kovács, Balázs; Massing, André (arXiv)
We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.

Viscoelastic Cahn–Hilliard models for tumor growth - Garcke, Harald; Kovács, Balázs; Trautwein, Dennis (Mathematical Models and Methods in Applied Sciences)
We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions [Formula: see text] by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula: see text], is required. Moreover, in arbitrary dimensions [Formula: see text], we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions [Formula: see text] and in two dimensions [Formula: see text], where a CFL condition is required. Then, in two dimensions [Formula: see text], we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions [Formula: see text].
Keywords: Applied Mathematics; Modeling and Simulation

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces - Elliott, Charles M.; Garcke, Harald; Kovács, Balázs (Numerische Mathematik)
Abstract An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $$H^1$$ H1 norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.
Keywords: Applied Mathematics; Computational Mathematics

Time-dependent electromagnetic scattering from thin layers - Nick, Jörg; Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Abstract The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems - Kovács, Balázs; Li, Buyang (IMA Journal of Numerical Analysis)
Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^∞ (0,T;W^{1,∞ })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^∞ (0,T;W^{1,∞ })$ norm is obtained by combining the stability analysis with the consistency estimates.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions - Altmann, Robert; Kovács, Balázs; Zimmer, Christoph (IMA Journal of Numerical Analysis)
Abstract This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $τ \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error estimates for a splitting integrator for abstract semilinear boundary coupled systems - Csomós, Petra; Farkas, Bálint; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces - Beschle, Cedric Aaron; Kovács, Balázs (Numerische Mathematik)
Abstract In this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics

FEM-BEM Coupling for the Maxwell–Landau–Lifshitz–Gilbert Equations via Convolution Quadrature: Weak Form and Numerical Approximation - Bohn, Jan; Feischl, Michael; Kovács, Balázs (Computational Methods in Applied Mathematics)
Abstract The full Maxwell equations in the unbounded three-dimensional space coupled to the Landau–Lifshitz–Gilbert equation serve as a well-tested model for ferromagnetic materials. We propose a weak formulation of the coupled system based on the boundary integral formulation of the exterior Maxwell equations. We show existence and partial uniqueness of a weak solution and propose a new numerical algorithm based on finite elements and boundary elements as spatial discretization with backward Euler and convolution quadrature for the time domain. This is the first numerical algorithm which is able to deal with the coupled system of Landau–Lifshitz–Gilbert equation and full Maxwell’s equations without any simplifications like quasi-static approximations (e.g. eddy current model) and without restrictions on the shape of the domain (e.g. convexity). We show well-posedness and convergence of the numerical algorithm under minimal assumptions on the regularity of the solution. This is particularly important as there are few regularity results available and one generally expects the solution to be non-smooth. Numerical experiments illustrate and expand on the theoretical results.
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis

A convergent finite element algorithm for mean curvature flow in higher codimension - Binz, Tim; Kovács, Balázs (arXiv)