The paper investigates the well-posedness of global solutions and the existence of global attractors for weakly damped FDS nonlinear wave equations. It establishes the well-posedness of weak solutions using Galerkin approximation and a priori estimate. Subsequently, a dynamical system is constructed based on the well-posedness of the solution. The existence of a bounded absorbing set for the equations and the smooth properties of the operator semigroup are presented, leading to the existence of a global attractor.

We develop a locally mass-conservative enriched Petrov-Galerkin (EPG) method without any penalty term for the simulation of Darcy flow in fractured porous media. The discrete fracture model is applied to model the fractures as the lower dimensional fracture interfaces. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions in the fractures and the surrounding porous media. We propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method. The solution of the pressure can be decoupled into two steps with a standard CG method and a post-processing correction. The post-processing correction based on the bubble functions in the matrix and the fractures can be solved separately, which is useful for parallel computing. We derive a priori and a posteriori error estimates for the problem. Numerical examples are presented to illustrate the performance of the proposed method.

For any positive integer $k,$ the reconfiguration graph for all $k$-colorings of a graph $G,$ denoted by $\mathcal{R}_k(G),$ is the graph where vertices represent the $k$-colorings of $G,$ and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any 2-chromatic $P_5$-free graph $G,$ $\mathcal{R}_k(G)$ is connected for each $k≥3.$ On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p,$ such that $\mathcal{R}_{8p}(G)$ is disconnected.
In this paper, we offer a detailed classification of the connectivity of $\mathcal{R}_k(G)$ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t = 3,$ and $t ≥ 4$ with $t+1 ≤ k ≤(\begin{matrix} t \\ 2 \end{matrix}).$ We demonstrate that $\mathcal{R}_k(G)$ remains connected for each 3-chromatic $P_5$-free graph $G$ and each $k ≥4.$ Furthermore, for each $t≥4$ and $t+1\le k\le (\begin{matrix} t \\ 2 \end{matrix}),$ we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel.

This paper proposes a robust and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method for solving multicomponent chemically reacting flows, which is an extension and application of the recent work [M. Peng, Z. Sun, and K. Wu, Math. Comput., 2024, doi.org/10.1090/mcom/3998]. Following recently developed high-order bound-preserving discontinuous Galerkin method in [J. Du and Y. Yang, J. Comput. Phys., 469 (2022), 111548], we incorporate an OE procedure after each Runge-Kutta time stage to suppress spurious oscillations. The OE procedure is defined by the solution operator of a damping equation, which can be analytically solved without requiring discretization, making its implementation straightforward, non-intrusive, and efficient. Through careful design of the damping coefficients, the proposed OEDG method not only achieves the essentially non-oscillatory (ENO) property without compromising accuracy but also preserves the conservative property — an indispensable aspect of the bound-preserving technique introduced in [J. Du and Y. Yang, J. Comput. Phys., 469 (2022), 111548]. The effectiveness and robustness of the OEDG method are demonstrated through a series of one- and two-dimensional numerical tests on the compressible Euler and Navier–Stokes equations for chemically reacting flows. These results highlight the method’s capability to handle complex flow dynamics while maintaining stability and high-order accuracy.