For numerical computation of three-dimensional (3-D) large-scale magnetostatic problems, iterative solver is preferable since a huge amount of memory is needed in case of using sparse direct solvers. In this paper, a recently proposed Coulomb-gauged magnetic vector potential (MVP) formulation for magnetostatic problems is adopted for finite element discretization using edge elements, where the resultant linear system is symmetric but ill-conditioned. To solve such linear systems efficiently, we exploit iterative Krylov subspace solvers by constructing three novel block preconditioners, which are derived from conventional block Jacobi, Gauss-Seidel and constraint preconditioners. Spectral properties and practical implementation details of the proposed preconditioners are also discussed. Then, numerical examples of practical simulations are presented to illustrate the efficiency and accuracy of the proposed methods.

When two overtaking bodies sail in close proximity, the additional sway force and yaw moment will be induced due to the changed relative velocity in the narrow clearance between them, which directly affect their course keeping. Up till now, the extensive data about the related body-to-body interaction effects are typically obtained from the captive model tests or the computational fluid dynamics (CFD) based calculations for the model fixed condition. In this work, the objective is to study the viscous hydrodynamic forces between two overtaking bodies for the model free condition. By coupling the incompressible Reynolds-averaged Navier-Stokes (RANS) equations and the motion equations of the two bodies, the complicated multi-degree of freedom motions in surge, sway and yaw are simulated. The detailed flow field characteristics, motion parameters including motion trajectories and yaw angles, and the viscous hydrodynamic forces acting on two bodies are procured. The effects of relative speed and various lateral distances are investigated to elucidate their influences on the interactional forces. The flow features at different time instants and the bodies' trajectories shown in the present article are helpful for the navigator to understand the mechanism of body-to-body collision accident.

In this paper, we propose an efficient numerical method for the optimal control problem constrained by elliptic equations. Being approximated by the finite element method (FEM), the continuous optimal control problem is discretized into a finite dimensional optimization problem with separable structures. Furthermore, an alternating direction method of multipliers (ADMM) is applied to solve the discretization problem. The total convergence analysis which includes the discretization error by FEM and iterative error by ADMM is established. Finally, numerical simulations are presented to verify the efficiency of the proposed method.

In this paper, we present a rigorous error analysis of a new higher order fractional-step scheme for approximation of the time-dependent Navier-Stokes equations. The main feature of the proposed scheme is twofold. First, it is a two-step scheme in which the incompressibility and nonlinearities are split. Second, this scheme is a linear scheme and is simple to implement. It is shown that the proposed scheme possesses the convergence rate $\mathcal O((\Delta t)^{3/2})$ in the discrete $l^2$(H$_0^1)\cap$ $l^\infty$(L$^2$)-norm for the end-of-step velocity. Two different numerical experiments are presented to confirm the theoretical analysis and the efficiency of the proposed scheme.

The Erdélyi-Kober fractional operators have found a large number of applications in many disciplines such as porous media and viscoelasticity. The purpose of this paper is to express the fractional integro-differential equations with Erdélyi-Kober derivative in terms of a class of nonlinear weakly singular integral equations of mixed type in order to analyze their numerical solvability. The resulting mixed type Volterra equations will have kernels containing both an end point and diagonal singularity, with solutions that their derivatives typically are unbounded. Applications of such problems are described to reformulate the fractional integro-differential equations with  Erdélyi-Kober derivative in terms of a particular class of cordial weakly singular integral equations of mixed type. The existence and uniqueness results of solutions under some verifiable conditions on the kernels and nonlinear functions are discussed. The corresponding nonlinear weakly singular equation can be solved numerically in terms of the implicitly linear collocation method. The error analysis of the method is also discussed and the feasibility of the introduced strategy is illustrated by some numerical experiments. The reformulation proposed here might be used to develop a computational method to solve fractional integro-differential equations.

This paper is concerned with two types of two-grid stabilized finite element methods (FEMs) based on Newton iteration for the steady-state nature convection problem. The first method needs to solve one small nonlinear natural convection system on the coarse mesh with mesh width $H$, and then to solve one large linearized natural convection system on the fine mesh with mesh width $h=\mathcal {O}(H^2)$ based on Newton iteration. The other method needs to solve one small nonlinear natural convection system on the same coarse mesh, and then to solve two large linearized systems on the fine mesh with mesh width $h=\mathcal {O}(H^{\frac{7-\varepsilon}{2}})$ based on Newton iteration which have the same stiffness matrix with only different right-hand side. In both methods, the stabilization terms are defined via two local Gauss integrations at element level which has no need to introduce additional variables comparing with the standard variational multiscale stabilized FEMs. The stability estimates and the convergence analysis for both methods are derived strictly. Ample numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the new methods.

High-order finite difference schemes (FDSs) based on symmetric conservative metric method (SCMM) are investigated. Firstly, the decomposition and geometric meaning of the discrete metrics and Jacobian based on SCMM are proposed. Then, high-order central FDS based on SCMM is proved to be a weighted summation of second-order finite difference schemes (FDSs). Each second-order FDS has the same vectorized surfaces and cell volume as a second-order finite volume scheme (FVS), and the cell volume is uniquely determined by the vectorized surfaces. Moreover, the decomposition and connection with FVSs are also discussed for general high-order FDSs. SCMM can be applied for high-order weighted compact nonlinear scheme (WCNS). Numerical experiments show superiority of high-order WCNS based on SCMM in stability, accuracy and ability to compute flows around complex geometries. The results in this paper may to some extent explain why high-order FDSs based on SCMM can solve problems with complex geometries and may give some guidance in constructing high-order FDSs on curvilinear coordinates.

In this paper, a Jacobi spectral collocation approximation is proposed for the solution of second-order Volterra integro-differential equations with weakly singular kernels. The solution of such equations usually exhibits a singular behaviour at the origin. By using some suitable variable transformations, we obtain a new equation which is still weakly singular, but whose solution is as smooth as we like. Then the resulting equation is solved by standard spectral methods. We establish a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. Finally, to perform the numerical simulation, a test example is considered with non-smooth solutions.

We present a numerical assessment of a class of criteria for mesh adaptation in the finite volume solution of shallow water flows. The shallow water equations are numerically approximated by a predictor-corrector procedure in unstructured triangular meshes. The numerical fluxes at the interfaces of each triangle are reconstructed in the predictor stage using an upwind scheme along with slope limiters to achieve a second-order accuracy. Treatment of source terms is performed in the corrector stage using a well-balanced technique. Four error indicators using the flow variables are discussed and applied as criteria for the mesh adaptation. Numerical results are presented for two test examples for a circular dam-break flow and dam-break problem over a single building. The presented criteria are found to give accurate results in comparison with similar simulations carried out using uniformly refined fixed meshes. Dynamic grid adaptation and the use of an explicit time integration scheme are found to enhance the computational efficiency of the finite volume solution of shallow water flows. In addition, the obtained results for dam-break problems are considered to be representative, and might be helpful for a fair rating of criteria for mesh adaptation in the finite volume solution of shallow water flows, particularly in long time computations.

In this paper, we investigate a mixed finite element method (MFEM) for the elliptic optimal control problems (OCPs) with a distributive control. The state variable and adjoint state variable are approximated by the conforming rectangular $Q_{11}+Q_{01}\times Q_{10}$ elements pair. The discrete B-B condition is satisfied automatically, which is usually considered to be the key point of the MFEM. The control is then obtained by the orthogonal projection through the adjoint state. Optimal orders of convergence are derived for the above mentioned variables. Furthermore, supercloseness and superconvergence results are also established under certain reasonable regularity assumptions. Some numerical results are provided to verify the theoretical analysis. At last, the proposed method is extended to some other low order conforming and nonconforming elements.

In this paper, a variable correction-based immersed boundary method (IBM) for simulating viscous compressible flows over stationary and moving bodies is presented. It is the extension of the implicit velocity correction-based IBM proposed by Wu and Shu (J. Comput. Phys., 228 (2009), pp.  1963-1979). Since the incompressible flow was studied in their work, only the fluid velocity field was needed to be corrected by enforcing the no-slip boundary condition. But for compressible flow problems, other flow variables around the boundary also should be corrected or updated. To simulate the flow field without the effect of immersed boundary firstly, the open source code OpenFOAM is utilized in this work. After performing the velocity correction then, the density is corrected by resolving the continuity equation with the corrected fluid velocity. After that, the temperature surrounding the boundary is corrected from the given temperature condition, which is similar to the velocity correction. Finally, the pressure is updated by using the corrected density and temperature through the equation of state. In such way, all the flow variables have been corrected and the given physical boundary conditions can be accurately implemented. To validate the proposed method, the flows over a stationary circular cylinder, a stationary airfoil and a transversely oscillating circular cylinder at different Mach and Reynolds numbers are simulated. Compared to the results in the literature, good agreement can be achieved. To further illustrate the potential of current method for dealing with complex problems, the flow over a rotating cylinder in front of a stationary cylinder is also simulated.

In this paper, we construct and analyze the a posteriori error estimators for any order finite volume methods (FVMs) for solving the elliptic boundary value problems in $R^2$. We shall prove that the a posteriori error estimators yield the global upper and local lower bounds for the $H^1$-norm error of the corresponding FVMs. So that the a posteriori error estimators are equivalent to the true errors in a certain sense. Lots of numerical experiments are performed to illustrate the theoretical results.

Retinex theory explains that the image intensity is the product of the object's reflectance and illumination. However, the true color of the object in the image is determined only by the reflectance of the object. The purpose of retinex problem is to decompose the reflectance from the image intensity. In this paper, a new variational model with physical constraint imposed on the reflectance is proposed. The proposed model can be transformed to a linear complementarity problem (LCP) with symmetric positive semi-definite (SPSD) matrix. The main contribution of the paper is that the LCP with SPSD matrix is solved by the modulus iteration method and the convergence is demonstrated. Experiments numerically show the effectiveness of the proposed method for retinex problem and the convergence of the modulus iteration method for solving the LCP with SPSD matrix.

This study develops a novel high-order three-scale (HOTS) computational method to accurately simulate and analyze the thermoelastic behaviors of axisymmetric composite structures with multiple spatial scales. The inhomogeneities in composite structures are taken into account by periodic distributions of representative unit cells on the mesoscale and microscale. Firstly, the multiscale asymptotic analysis for these multiscale problems is given in detail, and the new unified micro-meso-macro HOTS approximate solutions for these multiscale problems are established based on the above-mentioned multiscale analysis. Two types of auxiliary cell functions are established on mesoscale and microscale. Also, two kinds of equivalent material parameters are calculated by up-scaling procedure on the mesoscale and microscale, and the homogenized problems are subsequently defined on global structure. Then, the numerical accuracy analyses for the conventional two-scale solutions, low-order three-scale (LOTS) solutions and HOTS solutions are obtained in the pointwise sense. By the foregoing error analyses, the vital necessity of developing HOTS solutions for simulating these three-scale problems is illustrated clearly. Furthermore, the corresponding HOTS numerical algorithm based on finite element method (FEM) is brought forward in detail. Finally, some numerical examples are presented to verify the usability and effectiveness of the HOTS computational method developed in this work.