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This paper aims to propose two parallel two-step finite element algorithms based on fully overlapping domain decomposition for solving the 2D/3D time-dependent natural convection problem.

The first-order implicit Euler formula and second-order Crank–Nicolson formula are used to time discretization respectively. Each processor of the algorithms computes a stabilized solution in its own global composite mesh in parallel. These algorithms compute a nonlinear system for the velocity, pressure and temperature based on a lower-order element pair (P₁b-P₁-P₁) and solve a linear approximation based on a higher-order element pair (P₂-P₁-P₂) on the same mesh, which shows that the new algorithms have the same convergence rate as the two-step finite element methods. What is more, the stability analysis of the proposed algorithms is derived. Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms.

Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms.

The novel parallel two-step algorithms for incompressible natural convection problem are proposed. The rigorous analysis of the stability is given for the proposed parallel two-step algorithms. Extensive 2D/3D numerical tests demonstrate that the parallel two-step algorithms can deal with the incompressible natural convection problem for high Rayleigh number well.

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.

The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.

A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.

The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

Motivated by the acoustics of motor vehicles, a coupled fluid–solid system is considered. The air pressure is modeled by the Helmholtz equation, and the structure displacement is described by elastodynamic equations. The acoustic–structure interaction is modeled by coupling conditions on the common interface. First, the existence and uniqueness of solutions are investigated, and then, after recalling fundamental notions of shape optimization, the tensor form of the distributed shape derivative is obtained for the coupled problem. It is then applied to the minimization of the sound pressure by variation of the structure shape through the positioning of beads.

The existence and uniqueness of solutions up to eigenfrequencies are shown by the Fredholm–Riesz–Schauder theory using a novel decomposition into an isomorphism and a compact operator. For the design optimization, the distributed shape derivative is obtained using the averaged adjoint method. It is then used in a closed 3D optimization process of the position of a bead for noise reduction. In this process, the C++ library concepts are used to solve the differential equations on hexahedral meshes with the finite element method of higher order.

The existence and uniqueness of solutions have been shown for the case without absorption, where the given proof allows for extension to the case with absorption in the domain or via boundary conditions. The theoretical results show that the averaged adjoint can be applied to compute distributed shape derivatives in the context of acoustic–structure interaction. The numerical results show that the distributed shape derivative can be used to reduce the sound pressure at a chosen frequency via rigid motions of a nonsmooth shape.

The proof of shape differentiability and the calculation of the distributed shape derivative in tensor form allows to consider nonsmooth shapes for the optimization, which is particularly relevant for the optimal placement of beads or stampings in a structural-acoustic system.