Mathematical Models and Methods in Applied Sciences

In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a non-conforming spatial approximation and finite differences in time (Rothe–Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Therefore, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We formulate the problem in the very general and abstract setting of so-called non-conforming Bochner pseudo-monotone operators, which allows for a unified treatment of several evolution problems. Our abstract results for non-conforming Bochner pseudo-monotone operators allow to establish (weak) convergence just by verifying a few natural assumptions on the operators time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be performed easily. We exemplify the applicability of our approach on several DG schemes for the unsteady $p$-Navier–Stokes problem. The results of some numerical experiments are reported in the final section.

We consider the motion of a viscous compressible and heat conducting fluid confined in the gap between two rotating cylinders (Taylor–Couette flow). The temperature of the cylinders is fixed but not necessarily constant. We show that the problem admits a time-periodic solution as soon as the ratio of the angular velocities of the two cylinders is a rational number.

In this work, the concept of dual natural-norm for parametrized linear equations is used to derive residual-based a posteriori error bounds characterized by a ${𝒪}(1)$ stability constant. We translate these error bounds into very effective practical a posteriori error estimators for reduced basis approximations and show how they can be efficiently computed following an offline/ online strategy. We prove that our practical dual natural-norm error estimator outperforms the classical inf–sup based error estimators in the self-adjoint case. Our findings are illustrated on anisotropic Helmholtz equations showing resonant behavior. Numerical results suggest that the proposed error estimator is able to successfully catch the correct order of magnitude of the reduced basis approximation error, thus outperforming the classical inf–sup based error estimator even for non-self-adjoint problems.

In this paper, we develop stability analysis, including inverse inequality, $L^2$ norm equivalence and interpolation error estimates, for divergence free conforming virtual elements in arbitrary dimension. A local energy projector based on the local Stokes problem is suggested, which commutes with the divergence operator. After defining a discrete bilinear form and a stabilization involving only the boundary degrees of freedom (DoF) and parts of the interior DoF, a new divergence free conforming virtual element method is advanced for the Brinkman problem, which can be reduced to a simpler method due to the divergence free discrete velocity. An optimal convergence rate is derived for the discrete method. Furthermore, we achieve a uniform half convergence rate of the discrete method in consideration of the boundary layer phenomenon. Finally, some numerical results are provided to validate the convergence of the discrete method.

This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well.

Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves (TWs) move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction–diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic TWs move is an interval unbounded to the right. Contrary to classic examples, the infimum may not be reached. When the flow is elliptic or over-elliptic, the minimum speed of propagation is achieved. The classic TW speed threshold is complemented by another value by analyzing an extension of the first-order boundary value problem to which the classic case is reduced. This singular minimum speed can be justified as a viscous limit of classic minimal speeds in elliptic or over-elliptic flows. We construct a singular profile for each speed between the minimum singular speed and the speeds at which classic TWs move. Under additional assumptions, the constructed profile can be justified as that of a TW of the starting equation in the framework of bounded variation functions. We also show that saturated fronts verifying the Rankine–Hugoniot condition can appear for strictly lower speeds even in the framework of bounded variation functions.