In this paper, a new locking-free mixed full-order model (FOM) for solving the elastic wave equation is studied, and then a locking-free reduced-order model (ROM) based on the proper orthogonal decomposition (POD) technique is constructed, which greatly improves solving efficiency compared to FOM while maintaining the locking-free. Theoretical analysis of semi discrete and fully discrete schemes for the FOM and the ROM are also presented. Some numerical experiments verify the theoretical analysis results.

Utilizing the nonlinear Time Fractional Degenerate Parabolic Equation (TFDPE) in modeling provides a comprehensive approach to studying phenomena exhibiting both fractional order dynamics and degenerate parabolic behavior, facilitating accurate predictions and insights across diverse scientific domains. However, the numerical solution of TFDPE is a challenging task and traditional numerical methods cannot solve this equation because of the spatial singularity influence. In this paper, we find the numerical solution of nonlinear TFDPE with both strongly and weakly degenerate cases using the higher order augmented finite volume method on uniform grids. To handle the singularity of TFDPE, we choose an intermediate point near the singular point and split the whole domain into singular and regular subdomains. Then, we find the solution on singular subdomain using the Puiseux series while on the regular subdomain we find the solution by finite volume schemes. The main idea is to recover the Puiseux series on singular subdomain using the Picard iteration methods which is also a challenging because of the time fractional derivative in the original equation. The solution on the singular subdomain is in the form of Puiseux series, which has multiple undetermined augmented variables and these variables play a role in organically combining the singular and regular subdomains. To approximate the time fractional derivative, we use the second order weighted and shifted Grünwald difference (WSGD) scheme and give the comparison of our results with $L_1$-scheme. We use the discrete energy method to prove that the schemes has temporal second order while spatial second and fourth-order on the whole domain and for the augmented variables in discrete $L_2$-norm. Finally, we give some numerical examples to confirm the accuracy and order of convergence of the proposed schemes for the whole domain and the augmented variables. We also give an interesting example with coefficient blow-up at the degenerate point and show the schemes are working the same as the other cases.

In this article, we concentrate on the fast numerical computation of the radiation energy densities together with electron and ion temperatures of three-dimensional multi-group radiation diffusion equations, which is temporally discretized with the adaptive backward Eulerian scheme, linearized iteratively via the method of frozen coefficients and spatially approximated through a cell-centered finite volume discretization on the adaptive unstructured computational meshes. We present, analyze and implement an alternating positive semidefinite splitting preconditioning technique with two selective relaxations and algebraic multigrid subsolves, and provide an algebraic quasi-optimal selection approach to determine the involved parameters. Our parallel implementation is based on the software package jxpamg and the preconditioned flexible restarted generalized minimal residual solver has been examined by running realistic simulations of hydrodynamic instability on the Tianhe-2A supercomputer to demonstrate its numerical robustness, computational efficiency, parallel strong and weak scalabilities, and the competitiveness with some existing popular monolithic and block preconditioning strategies.

We develop a model for wind farm power optimization while considering the wake interaction among wind turbines. The proposed model is a Navier-Stokes equations-constrained optimization model with the objective of maximizing the total power where the operating points of the turbines are the decision variables, and the three-dimensional vorticity-velocity Navier-Stokes equations of wind speed are among the constraints. Moreover, we develop an efficient numerical algorithm to solve the optimization model. This algorithm is based on the pattern search method, the actuator line method and a time-stepping scheme which is used to solve the vorticity-velocity Navier-Stokes equations. In the numerical experiments, we first compute the power generation of a commercial wind turbine called WindSpot for different wind speed. It is shown that the computed power is in a good agreement with the measurements. Then, in the case of two turbines, we find that by optimizing the turbines’ operation while considering the wake effect, we can gain an additional 8.11% in the total power when the incoming wind speed on the boundary is 10 m/s.

We extend the cell-centered finite element method (CCFE) [1] with imposed flux continuity and streamline upwind technique to solve the advection-diffusion equations with anisotropic and heterogeneous diffusivity and a convection-dominated regime on general meshes. The scheme is cell-centered in the sense that the solution is computed by cell unknowns of the primal mesh. From general meshes, the method is constructed by the dual meshes and their triangular submeshes. The scheme gives auxiliary edge unknowns interpolated by the multipoint fluid approximation technique to obtain the local continuity of numerical fluxes across the interfaces. In addition, the scheme uses piecewise linear functions combined with a streamline upwind technique on the dual submesh in order to stabilize the numerical solutions and eliminate the spurious oscillations. The coercivity, the strong and dual consistency, and the convergence properties of this method are shown in the rigorous theoretical framework. Numerical results are carried out to highlight accuracy and computational cost.

The conforming discontinuous Galerkin (CDG) finite element method is an innovative and effective numerical approach to solve partial differential equations. The CDG method is based on the weak Galerkin (WG) finite element method, and removes the stabilizer in the numerical scheme. And the CDG method uses the average of the interior function to replace the value of the boundary function in the standard WG method. The integration by parts is used to construct the discrete weak gradient operator in the CDG method. This paper uses the CDG method to solve the parabolic equation. Firstly, the semi-discrete and full-discrete numerical schemes of the parabolic equation and the well-posedness of the numerical methods are presented. Then, the corresponding error equations for both numerical schemes are established, and the optimal order error estimates of $H^1$ and $L^2$ are provided, respectively. Finally, the numerical results of the CDG method are verified.