Guang-An Zou

In this paper, a new penalty-based least-squares discontinuous Galerkin (LS-DG) framework is proposed and analyzed for the time-dependent Stokes system. A key feature of the scheme is the use of a penalty method to address the incompressibility constraint, along with the introduction of an artificial average perturbation term in the weighted least-squares functional. This approach offers three distinct advantages: first, it facilitates the proof of coercivity of the functional, ensuring the unique solvability of the method; second, it guarantees the unconditional stability of the kinetic energy at the discrete level; and third, it integrates the strengths of both the discontinuous Galerkin and least-squares methods, naturally providing a built-in residual indicator for the adaptive procedure. We rigorously establish the well-posedness and derive optimal error estimates for the proposed scheme. Finally, the accuracy, energy stability, and high efficiency of the method are demonstrated through several numerical examples, which show that it outperforms the classical DG method in terms of computational efficiency.

In this article, we propose a novel numerical framework for the non‐isothermal Cahn–Hilliard–Navier–Stokes two‐phase flow system, which couples the incompressible Navier–Stokes equations, the Cahn–Hilliard phase‐field equation, and the heat transport equation to capture temperature‐dependent two‐phase flow dynamics. The proposed scheme achieves three major advances: (i) unconditional energy stability through a combined scalar auxiliary variable (SAV) and zero‐energy‐contribution (ZEC) approach, (ii) linearity and full decoupling of all variables while using a second‐order temporal discretization, and (iii) efficient implementation via discontinuous Galerkin (DG) spatial discretization together with a second‐order projection method for the Navier–Stokes equations. We rigorously prove the unconditional energy stability of the scheme and present key details of its decoupled implementation. Extensive 2D and 3D simulations, including droplet deformation, bubble coalescence, and interfacial instabilities in stratified binary fluids, are presented to demonstrate the accuracy, efficiency, and robustness of the proposed numerical method, thereby confirming its effectiveness for energy‐stable simulation of non‐isothermal two‐phase incompressible flows.

In this paper, we consider the data assimilation based on discrete-time observation. For ordinary differential equations, a new data assimilation algorithm, the discrete-time stochastic feedback control is introduced. Our main results show that the approximating solution, obtained by our algorithm, converges to the unknown reference solution over time. For two-dimensional Navier–Stokes equations, a discrete-time observation algorithm is also introduced. Our algorithm is designed to work with low Fourier modes and local spatial averages over finite volume elements. Numerical example is given to verify our results.

In this paper, we propose a new formulation of the artificial-viscosity penalty finite element methods for solving the mathematical model of non-Newtonian blood flow, treating blood as a pseudo-plastic fluid obeying the power-law Navier-Stokes equations. The proposed schemes are based on the artificial-viscosity mixed finite element method in the spatial direction, combined with a backward-Euler scheme and penalty-projection scheme for temporal discretization. It is rigorously derived the stability and error estimates for the fully discrete schemes. Numerical examples are presented to validate the theoretical ideas and to demonstrate the effectiveness of our proposed strategies. The comparison shows that the artificial viscosity penalty-projection scheme demands less central processing unit time, as well as a slightly higher-order precision. Finally, the penalty-projection finite element scheme is successfully applied to study the influence of blood viscosity on hemodynamics in aortic aneurysms. Moreover, numerical investigations for flow patterns based on patient-specific geometric models of cerebral aneurysms through segmentation of magnetic resonance imaging have been performed. The present study clearly illustrates the importance of taking the non-Newtonian properties of blood flow within aneurysms into account when studying the risk of aneurysm rupture.