In this paper, we propose a new method for the Darcy-Stokes equations based on the stabilizer-free weak Galerkin finite element method. In the proposed method, we remove the stabilizer term by increasing the degree of polynomial approximation space of the weak gradient operator. Compared with the classical weak Galerkin finite element method, it will not increase the size of global stiffness matrix. We show that the new algorithm not only has a simpler formula, but also reduces the computational complexity. Optimal order error estimates are established for the corresponding numerical approximation in various norms. Finally, we numerically illustrate the accuracy and convergence of this method.

In this paper, we prove the existence of weak solution and the uniqueness of strong solution to a Voigt-regularization of the three-dimensional thermally coupled inviscid, resistive MHD equations. We also propose a fully discrete scheme for the considered problem, which is proven to be stable and convergent. All computational results support the theoretical analysis and demonstrate the effectiveness of the presented scheme.

This article develops and analyses a mixed virtual element scheme for the spatial discretization of linear parabolic integro-differential equations (PIDEs) combined with backward Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection significantly helps in managing the integral terms, yielding optimal convergence of order $O(h^{k+1})$ for the two unknowns $p(x, t)$ and $\sigma(x, t).$ In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order $O(h^{k+2}).$ The fully discrete case has also been analyzed and discussed to achieve $O(\tau)$ in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions.

In the present work, we examine and analyze an $hp$-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system on computational meshes consisting of polytopic elements on the boundary. This approach is based on the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as has been introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and $H^1/L^2$-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes fluid system enabling the proof of the inf/sup condition and the $hp$- a priori error estimates, while we investigate the optimal convergence rates numerically. This approach recovers and integrates the flexibility and superiority of the discontinuous Galerkin methods for fluids whenever geometrical deformations are taking place by degenerating the edges, facets, of the polytopic elements only on the boundary, combined with the efficiency of the $hp$-version techniques based on arbitrarily shaped elements without requiring any mapping from a given reference frame.

In this paper, we rigorously analyze an HIV-1 infection model with CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model with proving the positivity and boundedness of the solution. For this model, the basic reproduction number $R_0$ and the immune reproduction number $R_1$ are identified. Moreover, we have shown that the model has three equilibria, namely the infection-free equilibrium $E_0,$ the infectious equilibrium without immune response $E_1$ and the infectious equilibrium with immune response $E_2.$ By applying fluctuation lemma and Lyapunov functionals, we have demonstrated that the global stability of $E_0$ and $E_1$ are only related to $R_0$ and $R_1.$ The local stability of the third equilibrium is obtained under four situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some numerical simulations are carried out for illustrating the theoretical results.

In this paper, we introduce a weak Galerkin finite element method for the dual-porosity-Stokes model. The dual-porosity-Stokes model couples the dual-porosity equations with the Stokes equations through four interface conditions. In this method, we define several weak Galerkin finite element spaces and weak differential operators. We provide the weak Galerkin scheme for the model, and establish the well-posedness of the numerical scheme. The optimal convergence orders of errors in the energy norm are derived. Finally, we verify the effectiveness of the numerical method with different weak Galerkin elements on different meshes.