Showing papers in "Mathematical Models and Methods in Applied Sciences in 2017"
TL;DR: This work analyzes the Virtual Element Methods on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature, and shows that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom.
Abstract: We analyze the virtual element methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter) can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the “classical” one introduced in Ref. 4, and a recent one presented in Ref. 34. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.
253 citations
TL;DR: In this paper, a set of first-and second-order temporal approximation schemes based on a novel "Invariant Energy Quadratization" approach is presented. But the scheme is not energy stable.
Abstract: How to develop efficient numerical schemes while preserving energy stability at the discrete level is challenging for the three-component Cahn–Hilliard phase-field model. In this paper, we develop a set of first- and second-order temporal approximation schemes based on a novel “Invariant Energy Quadratization” approach, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to well-posed linear systems with a linear symmetric, positive definite at each time step. We prove that the developed schemes are unconditionally energy stable and present various 2D and 3D numerical simulations to demonstrate the stability and the accuracy of the schemes.
171 citations
TL;DR: In this article, the authors present and discuss various one-dimensional linear Fokker-Planck type equations that have been recently considered in connection with the study of interacting multi-agent systems.
Abstract: We present and discuss various one-dimensional linear Fokker–Planck-type equations that have been recently considered in connection with the study of interacting multi-agent systems. In general, these Fokker–Planck equations describe the evolution in time of some probability density of the population of agents, typically the distribution of the personal wealth or of the personal opinion, and are mostly obtained by linear or bilinear kinetic models of Boltzmann type via some limit procedure. The main feature of these equations is the presence of variable diffusion, drift coefficients and boundaries, which introduce new challenging mathematical problems in the study of their long-time behavior.
115 citations
TL;DR: In this article, the authors considered the Keller-Segel-type parabolic system and showed that for all suitably regular initial data, the associated initial value problem possesses a globally defined bounded classical solution, provided that the motility function ϕ ∈ C3([0,∞)) ∩ W1, ∞(( 0,∆)) is uniformly positive.
Abstract: This work considers the Keller–Segel-type parabolic system ut = Δ(uϕ(v)), x ∈ Ω,t > 0, vt = Δv − v + u,x ∈ Ω,t > 0, (⋆) in a smoothly bounded convex domain Ω ⊂ ℝn, n ≥ 2, under no-flux boundary conditions, which has recently been proposed as a model for processes of stripe pattern formation via so-called “self-trapping” mechanisms In the two-dimensional case, in stark contrast to the classical Keller–Segel model in which large-data solutions may blow up in finite time, for all suitably regular initial data the associated initial value problem is seen to possess a globally-defined bounded classical solution, provided that the motility function ϕ ∈ C3([0,∞)) ∩ W1,∞((0,∞)) is uniformly positive In the corresponding higher-dimensional setting, it is shown that certain weak solutions exist globally, where in the particular three-dimensional case this solution actually is bounded and classical if the initial data are suitably small in the norm of L2(Ω) × W1,4(Ω) Finally, if still n = 3 but merely the physically interpretable quantity ∥ϕ′∥ L∞((0,∞))∫Ωu0 is appropriately small, then the above-weak solutions are proved to become eventually smooth and bounded
108 citations
TL;DR: This paper presents a kinetic equation incorporating the Cucker–Smale flocking force and stochastic game theoretic interactions in collision operators, and presents a sufficient framework leading to the asymptotic velocity alignment and global existence of smooth solutions for the proposed kinetic model with a special kernel.
Abstract: This paper addresses some preliminary steps toward the modeling and qualitative analysis of swarms viewed as living complex systems. The approach is based on the methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlocal, nonlinearly additive and modeled by theoretical tools of stochastic game theory. Collective learning theory can play an important role in the modeling approach. We present a kinetic equation incorporating the Cucker–Smale flocking force and stochastic game theoretic interactions in collision operators. We also present a sufficient framework leading to the asymptotic velocity alignment and global existence of smooth solutions for the proposed kinetic model with a special kernel. Analytic results on the global existence and flocking dynamics are presented, while the last part of the paper looks ahead to research perspectives.
101 citations
TL;DR: A novel first-order stochastic swarm intelligence model in the spirit of consensus formation models is introduced, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions.
Abstract: We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.
92 citations
TL;DR: In this paper, a Cahn-Hilliard-Darcy model is proposed to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions.
Abstract: We derive a Cahn–Hilliard–Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. A multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, are included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth are investigated numerically. In particular, for certain modelling choices, we obtain some form of focal and patchy necrotic growth that have been observed in experiments.
88 citations
TL;DR: A conceptual revisiting of population dynamics to include heterogeneous behaviors of individuals, mutations, and selection and the effect of the cellular aging in the virus infection process and the dynamics of virus mutation and competition with the immune system are proposed.
Abstract: This paper proposes a conceptual revisiting of population dynamics to include heterogeneous behaviors of individuals, mutations, and selection. The first part of the paper focuses on the derivation of a general mathematical structure which permits to describe systems composed of individuals whose interactions are stochastic. Hybrid models where some of the populations follow a deterministic dynamics are also discussed. The second part deals with two specific applications, namely the effect of the cellular aging in the virus infection process and the dynamics of virus mutation and competition with the immune system. Sample simulations are presented and classical models of population dynamics are critically analyzed in light of the proposed approach.
81 citations
TL;DR: In this paper, a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations was proposed to ensure exactly divergence-free approximations of the velocity and the magnetoreduction.
Abstract: We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magne...
80 citations
TL;DR: Porosity models and ST computational methods for compressible-flow aerodynamics of parachutes with geometric porosity are introduced and the main components of the ST computational framework the authors use are the compressibles-flow ST SUPG method, and the compressible -flow ST Slip Interface method, which is introduced here.
Abstract: Spacecraft-parachute designs quite often include “geometric porosity” created by the hundreds of gaps and slits that the flow goes through. Computational fluid–structure interaction (FSI) analysis of these parachutes with resolved geometric porosity would be exceedingly challenging, and therefore accurate modeling of the geometric porosity is essential for reliable FSI analysis. The space–time FSI (STFSI) method with the homogenized modeling of geometric porosity has proven to be reliable in computational analysis and design studies of Orion spacecraft parachutes in the incompressible-flow regime. Here we introduce porosity models and ST computational methods for compressible-flow aerodynamics of parachutes with geometric porosity. The main components of the ST computational framework we use are the compressible-flow ST SUPG method, which was introduced earlier, and the compressible-flow ST Slip Interface method, which we introduce here. The computations we present for a drogue parachute show the effectiv...
75 citations
TL;DR: In this article, a compressible Euler-type equation with singular commutator was derived from a hyperbolic limit of the kinetic description to the Cucker-Smale model of interacting individuals.
Abstract: This paper deals with the derivation and analysis of a compressible Euler-type equation with singular commutator, which is derived from a hyperbolic limit of the kinetic description to the Cucker–Smale model of interacting individuals.
TL;DR: In this paper, a weak solution concept for the Keller-Segel-Navier-Stokes system was developed, which requires solutions to satisfy very mild regularity hypotheses only for the component n.
Abstract: This paper deals with the Keller–Segel–Navier–Stokes system nt + u ⋅∇n = Δn −∇⋅ (nS(x,n,c) ⋅∇c), ct + u ⋅∇c = Δc − c + n, ut + (u ⋅∇)u = Δu −∇P + n∇ϕ,∇⋅ u = 0, in a bounded domain Ω ⊂ ℝ3 with smooth boundary, where ϕ ∈ W2,∞(Ω) and S ∈ C2(Ω × [0,∞)2; ℝ3×3) are given functions. We shall develop a weak solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component n. Under the assumption that there exist S0 > 0 and α > 1 3 such that |S(x,n,c)|≤ S0 ⋅ (1 + n)−αfor all x ∈Ω, n ≥ 0 and c ≥ 0, it is finally shown that for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined weak solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on α here is optimal. This result extends previous studies on global solvability for this system in the two-dimensional domain and for the associated chemotaxis-Stokes system obtained on neglecting the nonlinea...
TL;DR: This paper introduces and analyzes a mixed virtual element method (mixed-VEM) for the two-dimensional Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions, in which the only unknown is given by the pseudostress, whereas the velocity and pressure are computed via postprocessing formulae.
Abstract: In this paper, we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. More precisely, we employ a dual-mixed formulation in which the only unknown is given by the pseudostress, whereas the velocity and pressure are computed via postprocessing formulae. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, in order to define a calculable discrete bilinear form, whose continuous version involves deviatoric tensors, we propose two well-known alternatives for the local projector onto a suitable polynomial subspace, which allows the explicit integration of these terms. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Lax–Milgram lemma. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken ℍ(div)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.
TL;DR: In this paper, a new model for multi-agent dynamics where each agent is described by its position and body attitude is presented, where agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations.
Abstract: We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. In this manner, the body attitude is described by three orthonormal axes giving an element in SO(3) (rotation matrix). Agents try to coordinate their body attitudes with the ones of their neighbours. In the present paper, we give the Individual Based Model (particle model) for this dynamics and derive its corresponding kinetic and macroscopic equations. The work presented here is inspired by the Vicsek model and its study in [24]. This is a new model where collective motion is reached through body attitude coordination.
TL;DR: In this paper, the authors considered the Cahn-Hilliard-Oono equation with singular (e.g. logarithmic) potential in a bounded domain of ℝd, d ≤ 3.
Abstract: We consider the so-called Cahn–Hilliard–Oono equation with singular (e.g. logarithmic) potential in a bounded domain of ℝd, d ≤ 3. The equation is subject to an initial condition and Neumann homogeneous boundary conditions for the order parameter as well as for the chemical potential. However, contrary to the Cahn–Hilliard equation, the total mass might not be conserved. The existence of a global finite energy solution to such a problem was proven by Miranville and Temam. We first establish some regularization properties in finite time of the (unique) solution. Then, in dimension two, we prove the so-called strict separation property, namely, we show that any finite energy solution stays away from pure phases, uniformly with respect to the initial energy and the total mass. Taking advantage of these results, we study the long-time behavior of solutions. More precisely, we establish the existence of the global attractor in both two and three dimensions. Due to the strict separation property in dimension tw...
TL;DR: In this article, the authors show that the heat equation with homogeneous Dirichlet boundary conditions is controllable under a state-constrained controllability property, but only if the control time is large enough.
Abstract: The heat equation with homogeneous Dirichlet boundary conditions is well known to preserve nonnegativity. Besides, due to infinite velocity propagation, the heat equation is null-controllable within arbitrary small time, with controls supported in any arbitrarily open subset of the domain (or its boundary) where heat diffuses. The following question then arises naturally: can the heat dynamics be controlled from a positive initial steady-state to a positive final one, requiring that the state remains nonnegative along the controlled time-dependent trajectory? We show that this state-constrained controllability property can be achieved if the control time is large enough, but that it fails to be true in general if the control time is too short, thus showing the existence of a positive minimal controllability time. In other words, in spite of infinite velocity propagation, realizing controllability under the unilateral nonnegativity state constraint requires a positive minimal time. We establish similar results for unilateral control constraints. We give some explicit bounds on the minimal controllability time, first in 1D by using the sinusoidal spectral expansion of solutions, and then in multi-D on any bounded domain. We illustrate our results with numerical simulations, and we discuss similar issues for other control problems with various boundary conditions.
TL;DR: In this article, it was shown that for time-harmonic electromagnetic waves scattered by either perfectly conducting or dielectric bounded obstacles, the fields depend holomorphically on the shape of the scatterer.
Abstract: For time-harmonic electromagnetic waves scattered by either perfectly conducting or dielectric bounded obstacles, we show that the fields depend holomorphically on the shape of the scatterer. In th...
TL;DR: A residual-based a posteriori error estimate for the Poisson problem with discontinuous diffusivity coefficient is derived in the case of a virtual element discretization using a computable residual based error estimator that does not involve any term related to the virtual element stabilization.
Abstract: A residual-based a posteriori error estimate for the Poisson problem with discontinuous diffusivity coefficient is derived in the case of a virtual element discretization. The error is measured considering a suitable polynomial projection of the discrete solution to prove an equivalence between the defined error and a computable residual based error estimator that does not involve any term related to the virtual element stabilization. Numerical results display a very good behavior of the ratio between the error and the error estimator, resulting independent of the meshsize and element distortion.
TL;DR: It is proved that the adaptive isogeometric method is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class.
Abstract: We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and continue the study of its numerical properties We prove that our AIGM is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class The main tool we use is the theory of adaptive methods, together with a local upper bound for the residual error indicators based on suitable properties of a well selected quasi-interpolation operator on hierarchical spline spaces
TL;DR: In this article, the authors considered time-discrete BV-evolutions for a phase-field model with fracture and damage, and characterized their time-continuous limit in terms of parametrized BVevolutions, introducing a suitable family of energy norms.
Abstract: We consider time-discrete evolutions for a phase-field model (for fracture and damage) obtained by alternate minimization schemes First, we characterize their time-continuous limit in terms of parametrized BV-evolutions, introducing a suitable family of “intrinsic energy norms” Further, we show that the limit evolution satisfies Griffith’s criterion, for a phase-field energy release, and that the irreversibility constraint is thermodynamically consistent
TL;DR: In this article, a variational approach to the learnability of non-local interaction kernels is presented, where the kernel to be learned is bounded and locally Lipschitz continuous and the initial conditions of the systems are drawn identically and independently at random according to a given initial probability distribution.
Abstract: In this paper, we are concerned with the learnability of nonlocal interaction kernels for first-order systems modeling certain social interactions, from observations of realizations of their dynamics. This paper is the first of a series on learnability of nonlocal interaction kernels and presents a variational approach to the problem. In particular, we assume here that the kernel to be learned is bounded and locally Lipschitz continuous and that the initial conditions of the systems are drawn identically and independently at random according to a given initial probability distribution. Then the minimization over a rather arbitrary sequence of (finite-dimensional) subspaces of a least square functional measuring the discrepancy from observed trajectories produces uniform approximations to the kernel on compact sets. The convergence result is obtained by combining mean-field limits, transport methods, and a Γ-convergence argument. A crucial condition for the learnability is a certain coercivity property of the least square functional, defined by the majorization of an L2-norm discrepancy to the kernel with respect to a probability measure, depending on the given initial probability distribution by suitable push forwards and transport maps. We illustrate the convergence result by means of several numerical experiments.
TL;DR: In this paper, a Hilbert type method was developed to derive models at the macroscopic scale for large systems of several interacting living entities whose statistical dynamics at the microscopic scale is delivered by kinetic theory methods.
Abstract: This paper develops a Hilbert type method to derive models at the macroscopic scale for large systems of several interacting living entities whose statistical dynamics at the microscopic scale is delivered by kinetic theory methods. The presentation is in three steps, where the first one presents the structures of the kinetic theory approach used toward the aforementioned analysis; the second step presents the mathematical method; while the third step provides a number of specific applications. The approach is focused on a simple system and with a binary mixture, where different time-space scalings are used. Namely, parabolic, hyperbolic, and mixed in the case of a mixture.
TL;DR: A multiscale model for tumor cell migration in a tissue network that proves the global existence of a solution and performs numerical simulations to illustrate its behavior, paying particular attention to the influence of the supplementary structure and of the adhesion.
Abstract: We propose a multiscale model for tumor cell migration in a tissue network. The system of equations involves a structured population model for the tumor cell density, which besides time and
position depends on a further variable characterizing the cellular state with respect to the amount
of receptors bound to soluble and insoluble ligands. Moreover, this equation features pH-taxis and
adhesion, along with an integral term describing proliferation conditioned by receptor binding. The
interaction of tumor cells with their surroundings calls for two more equations for the evolution of
tissue fibers and acidity (expressed via concentration of extracellular protons), respectively. The
resulting ODE-PDE system is highly nonlinear. We prove the global existence of a solution and
perform numerical simulations to illustrate its behavior, paying particular attention to the influence
of the supplementary structure and of the adhesion.
TL;DR: In this paper, the Dirichlet problem of a one-velocity viscous drift-flux model was considered and the existence of global bounded weak solutions was shown under weak assumptions on the initial data, which can involve transition to pure single-phase points or regions.
Abstract: In this paper, we consider the Dirichlet problem of a one-velocity viscous drift-flux model. One of the phases is compressible, the other one is weakly compressible. Under weak assumptions on the initial data, which can be discontinuous and large as well as involve transition to pure single-phase points or regions, we show existence of global bounded weak solutions. One main ingredient is that we employ a decomposition of the pressure term appearing in the mixture momentum equation into two components, one for each of the two phases. This paves the way for deriving a basic energy equality. In particular, upper bounds on the masses are extracted from the estimates provided by the energy equality. By relying on weak compactness tools we obtain an existence result within the class of weak solutions. An essential novel aspect of this analysis, compared to previous works on the same model, is that the solution space is large enough to allow for transition to single-phase flow without any constraints. In particular, one of the phases can vanish in a point while the other phase can persist. The key to achieve this result, which represents a major step forward compared to previous results for this model, is that we do not rely on any higher-order (i.e. derivatives in space) estimates on the masses or pressure, only low-order estimates provided by the energy equality and the uniform upper bounds on the liquid and gas mass.
TL;DR: Deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data, are proposed and analyzed.
Abstract: We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data...
TL;DR: In this paper, the authors derived a new form of thermodynamically consistent quasi-incompressible diffuse-interface Navier-Stokes-Cahn-Hilliard model for a two-phase flow of incompressible fluids with different densities.
Abstract: While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier–Stokes–Cahn–Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman–Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with non-solenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and breakup dynamics of droplets including pinching due to gravity.
TL;DR: Bertozzi et al. as mentioned in this paper developed two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion, and proposed an interface condition via continuity to connect these two regimes.
Abstract: In this paper, we develop two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion [A. Bertozzi, J. Rosado, M. Short and L. Wang, Contagion shocks in one dimension, J. Stat. Phys. 158 (2014) 647–664]. In the continuum limit, the mesoscopic kinetic equation produces a natural Eulerian limit with nonlocal interactions. However, such limit ceases to be valid when the underlying microscopic particle characteristics cross, corresponding to the blow up of the solution in the Eulerian system. One method is to couple these two situations — using Eulerian dynamics for regions without characteristic crossing and kinetic evolution for regions with characteristic crossing. For such a hybrid setting, we provide a regime indicator based on the macroscopic density and fear level, and propose an interface condition via continuity to connect these two regimes. The other method is based on a level set formulation for the continuum system. The level set equation shares similar forms as the kinetic equation, and it successfully captures the multi-valued solution in velocity, which implies that the multi-valued solution other than the viscosity solution should be the physically relevant ones for the continuum system. Numerical examples are presented to show the efficiency of these new methods.
TL;DR: In this article, a new flux limiter is proposed to make the algebraic flux correction finite element scheme linearity and positivity preserving on general simplicial meshes, and a precise definition of it is proposed and analyzed.
Abstract: This work is devoted to the proposal of a new flux limiter that makes the algebraic flux correction finite element scheme linearity and positivity preserving on general simplicial meshes. Minimal assumptions on the limiter are given in order to guarantee the validity of the discrete maximum principle, and then a precise definition of it is proposed and analyzed. Numerical results for convection–diffusion problems confirm the theory.
TL;DR: In this article, the authors proved an optimal Ws,p-approximation for elliptic projectors on local polynomial spaces, based on the classical Dupont-Scott approximation.
Abstract: In this work, we prove optimal Ws,p-approximation estimates (with p ∈ [1, +∞]) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scott approximation theor...
TL;DR: An adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension $d\ge2$.
Abstract: We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension d ≥ 2. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.