Explore the vast range of titles available.

In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Grünwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved; second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time are proved in detail, which is higher than the L1 approximation result with O ( τ 2 - α + τ 2 - β ) . Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU time is compared to verify our theoretical results.

In this paper, based on a two-grid method and a recent local and parallel finite element method, a parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem is proposed and analyzed. This method ensures that all the local subproblems on the fine grid can be solved in parallel. Optimal error bounds of the approximate solution are obtained. Finally, numerical experiments are presented to demonstrate the accuracy and effectiveness of the proposed method.

In order to reduce the loss of laser point cloud appearance contours by point cloud lightweighting, this paper takes the laser point cloud data of the segment beam of the expressway viaduct as a sample. After comparing the downsampling algorithm from many aspects and angles, the voxel grid method is selected as the basic theory of the research. By combining the characteristics of the normal vector data of the laser point cloud, the top surface point cloud edge data are extracted and the voxel grid method is fused to establish an optimized point cloud lightweighting algorithm. The research in this paper shows that the voxel grid method performs better than the furthest point sampling method and the curvature downsampling method in retaining the top surface data, reducing the calculation time and optimizing the edge contour. Moreover, the average offset of the geometric contour is reduced from 2.235 mm to 0.664 mm by the edge-optimized voxel grid method, which has a higher retention. In summary, the edge-optimized voxel grid method has a better effect than the existing methods in point cloud lightweighting.

Based on overset grid method, the problems about flow past circular cylinder were numerically studied, which include the flow around a single circular cylinder at Reynolds number Re=100 and Reynolds number Re=200, the flow past two tandem circular cylinders at Reynolds number Re=200 and dimensionless distance g * =4, and the flow around a lateral oscillating circular cylinder at Reynolds number Re=200. Overset grid method is effective in dealing with the problems of flow over multi-body and flow with moving boundaries. This grid method couple overlapping regions in an arbitrary manner through flow field information updating over acceptor cells in one region suing its donor cells in another region. The computed drag coefficient and lift coefficients of the circular cylinder, vortical structure in the flow wake and vortex shedding P+S mode are agreed well with the results in previous experiment investigations and numerical simulations.

This study aimed to explore the psychological effects of wooden frames and interiors in open-plan offices (OPO) and to understand the evaluation structure associated with wooden materials. A combination of quantitative subjective evaluations and qualitative interviews was conducted, utilizing the Evaluation Grid Method (EGM) to understand participants’ evaluation structure and criteria for wooden OPO. The study involved 20 Japanese participants who evaluated 16 computer-generated images of OPO, designed with varying factors related to wood placement, material, and spatial elements. Findings revealed that OPO utilizing wooden columns and beams significantly enhances users’ willingness to use the office compared to those without. Notably, the combination of wood’s placement and other spatial elements is critical rather than the mere quantity of wood. Furthermore, the study reveals that factors such as age and gender affect the willingness to use OPO suggesting that different user attributes may lead to varying perceptions of comfort and privacy. The results indicated that while the evaluation criteria for assessing spaces were generally shared across participants, the relative importance of specific items varied depending on user attributes. This study offers valuable insights into the complex evaluation structures related to wooden office spaces, contributing to the growing field of research on wooden interiors.

Based on two-grid discretizations, this paper introduces a parallel finite element method for the 2D/3D Navier–Stokes equations with damping. In this method, we first solve a fully nonlinear problem on a global coarse grid, and then solve linearized residual subproblems in overlapping fine grid subdomains to update the coarse grid solution by some local and parallel procedures. With the help of local a priori estimate for the finite element solution, errors of the approximate solution from the proposed method are estimated. Performance of the proposed method is also illustrated by some numerical tests.

In this paper, by combining the two-grid method with a partition of unity, a local and parallel partition of unity scheme is designed and investigated for the mixed Navier-Stokes-Darcy problem. The main features of the present method are the following: (1) Once a coarse solution is derived, both the linearized Navier-Stokes and Darcy subproblems with a finer grid are independent, which allows a parallel computing with little communication; (2) a partition of unity is considered to assemble the solutions obtained from the local subdomains to arrive at a global continuous approximation; (3) a further coarse grid correction is carried out to derive optimal error bounds for the fluid velocity and piezometric head in L 2 -norm. Moreover, the convergence of the proposed method is shown. Some numerical experiments are reported to demonstrate the theoretical results.

A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium by employing a mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics (CEMFEM) for the concentration equation. The key point is to use a two-grid scheme to linearize the nonlinear term in the coupling equations. The main procedure of the algorithm is to solve small scaled nonlinear equations on the coarse grid and to deal with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Then, it is shown that a two-grid algorithm achieves optimal approximation as long as the mesh sizes satisfy H = O ( h 1 2 ) . Finally, numerical experiments confirmed the numerical analysis of two-grid algorithm.

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size and the fine grid size satisfy ( ), where is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

In this work, we propose and analyze a high-order mapping operator between two grids to construct a high-order two-grid difference algorithm for nonlinear partial differential equations. This algorithm is then applied to solve a nonlinear time-fractional biharmonic equation for illustration, in which the cut-off technique and the auxiliary scheme approach are used to reduce the requirement on the nonlinear term to the local Lipschitz continuous condition, and the energy estimates are performed to avoid the usage of the inverse estimates and thus the time-step conditions. To treat the fractional operator, the Alikhanov’s scheme on the graded mesh is applied to deal with the weak singularity at initial time, while the compact difference method based on the order reduction is employed for high-order spatial discretization. The above methods and improvements as well as the properties of the high-order mapping operator are integrated with the analysis of the two-grid method to prove the unique solvability and the unconditionally robust error estimates of the proposed schemes under different norms. The developed techniques are further extended to discretize and analyze the two-grid method of two-dimensional problems. Numerical examples are provided to verify the effectiveness and efficiency of the two-grid algorithms.