Error Estimates of Mixed Methods for Optimal Control Problems Governed by General Elliptic Equations

In this paper, we investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive $L^2$ and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.     

Superconvergence of Mixed Methods for Optimal Control Problems Governed by Parabolic Equations

In this paper, we investigate the superconvergence results for optimal control problems governed by parabolic equations with semi-discrete mixed finite element approximation. We use the lowest order mixed finite element spaces to discrete the state and costate variables while use piecewise constant function to discrete the control variable. Superconvergence estimates for both the state variable and its gradient variable are obtained.     

A Posteriori Error Estimates of Triangular Mixed Finite Element Methods for Semilinear Optimal Control Problems

In this paper, we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods. The state and co-state are approximated by the order $k\leq 1$ Raviart- Thomas mixed finite element spaces and the control is approximated by piecewise constant element. We derive a posteriori error estimates for the coupled state and control approximations. A numerical example is presented in confirmation of the theory.     

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

In this paper, we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions. We then set up its weak formulation and the finite element approximation scheme. Based on these we derive the a priori error estimates for its finite element approximation both in $H^1$ and $L^2$ norms. Furthermore, some numerical tests are presented to verify the theoretical results.     

Finite Element $θ$-Schemes for the Acoustic Wave Equation

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.     

Superconvergence of bi-$k$ Degree Time-Space Fully Discontinuous Finite Element for First-Order Hyperbolic Equations

In this paper, we present a superconvergence result for the bi-$k$ degree time-space fully discontinuous finite element of first-order hyperbolic problems. Based on the element orthogonality analysis (EOA), we first obtain the optimal convergence order of discontinuous Galerkin finite element solution. Then we use orthogonality correction technique to prove a superconvergence result at right Radau points, which is one order higherthan the optimal convergence rate. Finally, numerical results are presented to illustrate the theoretical analysis.     

Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials

Numerical simulation of metamaterials has attracted more and more attention since 2000, after the first metamaterial with negative refraction index was successfully constructed. In this paper we construct a fully-discrete leap-frog type finite element scheme to solve the three-dimensional time-dependent Maxwell’s equations when metamaterials are involved. First, we obtain some superclose results between the interpolations of the analytical solutions and finite element solutions obtained using arbitrary orders of Raviart–Thomas–Nédélec mixed spaces on regular cubic meshes. Then we prove the superconvergence result in the discrete l₂ norm achieved for the lowest-order Raviart–Thomas–Nédélec space. To our best knowledge, such superconvergence results have never been obtained elsewhere. Finally, we implement the leap-frog scheme and present numerical results justifying our theoretical analysis.     

Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$ which does not satisfy the inf-sup condition. The two-scale method consists of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the $H^1$-norm for velocity and the $L^2$-norm for pressure is obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation $h =\mathcal{O}(H^2)$. Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.     

Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the $H^1$ norm and the pressure in the $L^2$ norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.     

Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size $H$ and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size $h$. The error estimates derived show that if we choose $h=\mathcal{O}(|\log h|^{1/2}H^3)$, then the two-level method we provide has the same $H^1$ and $L^2$ convergence orders of the velocity and the pressure as the one-level stabilized method. However, the $L^2$ convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.     

Error Analysis and Adaptive Methods of Least Squares Nonconforming Finite Element for the Transport Equations

In this paper, we consider a least squares nonconforming finite element of low order for solving the transport equations. We give a detailed overview on the stability and the convergence properties of our considered methods in the stability norm. Moreover, we derive residual type a posteriori error estimates for the least squares nonconforming finite element methods under $H^{−1}$-norm, which can be used as the error indicators to guide the mesh refinement procedure in the adaptive finite element method. The theoretical results are supported by a series of numerical experiments.     

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.     

A Priori Error Estimates of Crank-Nicolson Finite Volume Element Method for Parabolic Optimal Control Problems

In this paper, the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation. The optimal convergent order $\mathcal{O}(h^2+k^2)$ is obtained for the numerical solution in a discrete $L^2$-norm. A numerical experiment is presented to test the theoretical result.     

On achievable rate of massive MIMO multiple access channels via virtual representation

This paper investigates the uplink achievable rates of massive multiple-input multiple-output (MIMO) systems in correlated fading channels via virtual representation. The fast fading MIMO channel matrix is assumed to have a Rayleigh-distributed random component with variance profile. Under the minimum mean-squared error receiver employed, we first derive the first and second asymptotic moments of signal-to-interference-plus-noise ratio (SINR). Then, we propose that the probability distribution function of SINR, which can be well approximated by a Gamma distribution. Finally, we derive a lower bound on the SINR and approximation of achievable rate. Numerical results demonstrate that both the lower bound on the SINR and the approximated rate apply for a finite number of antennas and remain tight.     

A posteriori error estimation and adaptive mesh refinement for a multiscale operator decomposition approach to fluid–solid heat transfer

We analyze a multiscale operator decomposition finite element method for a conjugate heat transfer problem consisting of a fluid and a solid coupled through a common boundary. We derive accurate a posteriori error estimates that account for all sources of error, and in particular the transfer of error between fluid and solid domains. We use these estimates to guide adaptive mesh refinement. In addition, we provide compelling numerical evidence that the order of convergence of the operator decomposition method is limited by the accuracy of the transferred gradient information, and adapt a so-called boundary flux recovery method developed for elliptic problems in order to regain the optimal order of accuracy in an efficient manner. In an appendix, we provide an argument that explains the numerical results provided sufficient smoothness is assumed.     

Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in $L^2$-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.     

The ultraviolet behaviour of integrable quantum field theories,affine Toda field theory

We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda field theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for theories in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against the numerical solutions. We address the question of existence and uniqueness of the solutions of the TBA equations, derive precise error estimates and determine the rate of convergence for the applied numerical procedure. A general expression for the Fourier transformed kernels of the TBA equations allows one to derive the related Y-systems and a reformulation of the equations into a universal form.     

油水两相问题的混合有限元逼近和特征线离散格式

本文讨论了多孔介质中油水两相驱动的一种有效方法。我们用混合有限元方法逼近压力和流速,而用特征线有限元方法逼近饱和度。在离散时间格式里,对压力-流速方程用较大的时间步长,而对饱和度方程用较小的步长,并使馆和度的矩阵分解数目在每层压力上降为1,这样大大减少了运算量。在对参数合理的限制下,我们得到了最佳收敛阶。     

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

Asymptotic Expansions and Extrapolations of $H^1$-Galerkin Mixed Finite Element Method for Strongly Damped Wave Equation

In this paper, a high-accuracy $H^1$-Galerkin mixed finite element method (MFEM) for strongly damped wave equation is studied by linear triangular finite element. By constructing a suitable extrapolation scheme, the convergence rates can be improved from $\mathcal{O}(h)$ to $\mathcal{O}(h^3)$ both for the original variable $u$ in $H^1(Ω)$ norm and for the actual stress variable $\boldsymbol{P}=∇u_t$ in $H$(div;$Ω$) norm, respectively. Finally, numerical results are presented to confirm the validity of the theoretical analysis and excellent performance of the proposed method.