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Hierarchical clustering in mean-field coupled Stuart--Landau oscillators
No full textClustered solutions in oscillator networks provide an important insight into how a system might diversify from a synchronous solution into spatiotemporal complex solutions. They can therefore form a link between fully synchronized and incoherent states. Despite their fundamental role in coupled oscillator dynamics, our understanding of how these clusters form and differentiate is still quite limited. Here, we study an ensemble of globally coupled Stuart--Landau oscillators and focus on the question of how 3-cluster solutions emerge from 2-cluster solutions and how the different 3-cluster solutions are organized in parameter space. We show that the arrangement of the clusters is dictated by a co-dimension 2 point, which we coin Type-II cluster singularity. Furthermore, our study points to a hierarchical structure of higher cluster solutions
On time-splitting methods for gradient flows with two dissipation mechanisms
No full textWe consider generalized gradient systems in Banach spaces whose evolutions are generated by the interplay between an energy functional and a dissipation potential. We focus on the case in which the dual dissipation potential is given by a sum of two functionals and show that solutions of the associated gradient-flow evolution equation with combined dissipation can be constructed by a split-step method, i.e. by solving alternately the gradient systems featuring only one of the dissipation potentials and concatenating the corresponding trajectories. Thereby the construction of solutions is provided either by semiflows, on the time-continuous level, or by using Alternating Minimizing Movements in the time-discrete setting. In both cases the convergence analysis relies on the energy-dissipation principle for gradient systems
Subdifferentials and penalty approximations of the obstacle problem
Full text linkWe consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalised problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving for example the positive part function max(0, ·). Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem
Variational modelling of porosity waves
Full text linkMathematical models for finite-strain poroelasticity in an Eulerian formulation are studied by constructing their energy-variational structure, which gives rise to a class of saddle-point problems. This problem is discretised using an incremental time-stepping scheme and a mixed finite element approach, resulting in a monolithic, structure-preserving discretisation. The Eulerian formulation is based on the inverse deformation, the so-called emphreference map. We present examples from geophysical applications, where elasticity and diffusive fluid flow are fully coupled and can be used to describe porosity waves, emphi.e., localised ascending fluid waves driven by gravitational forces
On the optimal control of viscous Cahn--Hilliard systems with hyperbolic relaxation of the chemical potential
No full textIn this paper, we study an optimal control problem for a viscous Cahn--Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence and regularity are known. We show Fréchet differentiability of the associated control-to-state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero
Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits
Full text linkWe study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on the case where the dynamics are driven by an entropy functional defined both on the metric edges and vertices, we provide a rigorous understanding of such systems of coupled ordinary and partial differential equations as (generalized) gradient flows in continuity equation format. Approximating the edges by a sequence of vertices, which yields a fully discrete system, we are able to establish existence of solutions in this formalism. Furthermore, we study several scaling limits using the recently developed framework of EDP convergence with embeddings to rigorously show convergence to gradient flows on reduced metric and combinatorial graphs. Finally, numerical studies confirm our theoretical findings and provide additional insights into the dynamics under rescaling
Discrete Transparent Boundary Conditions for Multi-Band Effective Mass Approximations
No full textThis chapter is concerned with the derivation and numerical testing of discrete transparent boundary conditions (DTBCs) for stationary multi-band effective mass approximations (MEMAs). We analyze the continuous problem and introduce transparent boundary conditions (TBCs). The discretization of the differential equations is done with the help of finite difference schemes.A fully discrete approach is used in order to develop DTBCs that are completely reflection-free. The analytical and discrete dispersion relations are analyzed in depth and the limitations of the numerical computations are shown. We extend the results of earlier works on DTBCs for the scalar Schrödinger equation by considering alternative finite difference schemes.The introduced schemes and their corresponding DTBCs are tested numerically on an example with a single barrier potential. The d-band k⋅p-model is introduced as most general MEMA. We derive DTBCs for the d-band k⋅p-model and test our results on a quantum well nanostructure
Quasi-optimal complexity hp-FEM for the Poisson equation on a rectangle
Full text linkWe show, in one dimension, that an hp-Finite Element Method (hp-FEM) discretisation can be solved in optimal complexity because the discretisation has a special sparsity structure that ensures that the reverse Cholesky factorisation---Cholesky starting from the bottom right instead of the top left---remains sparse. Moreover, computing and inverting the factorisation may parallelise across different elements. By incorporating this approach into an Alternating Direction Implicit (ADI) method {\`a} la Fortunato and Townsend (2020) we can solve, within a prescribed tolerance, an hp-FEM discretisation of the (screened) Poisson equation on a rectangle with quasi-optimal complexity: O(N^2 log N) operations where N is the maximal total degrees of freedom in each dimension. When combined with fast Legendre transforms we can also solve nonlinear time-evolution partial differential equations in a quasi-optimal complexity of O(N^2 log^2 N) operations, which we demonstrate on the (viscid) Burgers' equation. We also demonstrate how the solver can be used as an effective preconditioner for PDEs with variable coefficients, including coefficients that support a singularity
Derivation of the Reissner--Mindlin model from nonlinear elasticity
Full text linkWe discuss how the Reissner--Mindlin plate model can be derived from three-dimensional finite elasticity in terms of Gamma-convergence. The presence of transverse shear effects in the Reissner--Mindlin model requires to scale different components of the three-dimensional elastic strain differently. A main technical tool is then the combination of rigidity estimates for the deformation and suitably averaged versions
Approximation of time-periodic flow past a translating body by flows in bounded domains
Full text linkWe consider a time-periodic incompressible three-dimensional Navier-Stokes flow past a translating rigid body. In the first part of the paper, we establish the existence and uniqueness of strong solutions in the exterior domain that satisfy pointwise estimates for both the velocity and pressure. The fundamental solution of the time-periodic Oseen equations plays a central role in obtaining these estimates. The second part focuses on approximating this exterior flow within truncated domains, incorporating appropriate artificial boundary conditions. For these bounded domain problems, we prove the existence and uniqueness of weak solutions. Finally, we estimate the error in the velocity component as a function of the truncation radius, showing that, as the latter passes to infinity, the velocities of the truncated problems converge, in an appropriate norm, to the velocity of the exterior flow