Adaptive Finite Element Methods for Microstructures? Numerical Experiments for a 2-Well Benchmark

Macroscopic simulations of non-convex minimisation problems with enforced microstructures encounter oscillations on finest length scales – too fine to be fully resolved. The numerical analysis must rely on an essentially equivalent relaxed mathematical model. The paper addresses a prototype example, the scalar 2-well minimisation problem and its convexification and introduces a benchmark problem with a known (generalised) solution. For this benchmark, the stress error is studied empirically to asses the performance of adaptive finite element methods for the relaxed and the original minimisation problem. Despite the theoretical reliability-efficiency gap for the relaxed problem, numerical evidence supports that adaptive mesh-refining algorithms generate efficient triangulations and improve the experimental convergence rates optimally. Moreover, the averaging error estimators perform surprisingly accurate.     

A Rectangular Finite Volume Element Method for a Semilinear Elliptic Equation

In this paper we extend the idea of interpolated coefficients for semilinear problems to the finite volume element method based on rectangular partition. At first we introduce bilinear finite volume element method with interpolated coefficients for a boundary value problem of semilinear elliptic equation. Next we derive convergence estimate in H ¹-norm and superconvergence of derivative. Finally an example is given to illustrate the effectiveness of the proposed method. This work is supported by Program for New Century Excellent Talents in University of China State Education Ministry, National Science Foundation of China, the National Basic Research Program under the Grant (2005CB321703), the key project of China State Education Ministry (204098), Scientific Research Fund of Hunan Provincial Education Department, China Postdoctoral Science Foundation (No. 20060390894) and China Postdoctoral Science Foundation (No. 20060390894).     

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard \(L^2\)-norm and broken \(H^1\)-norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.     

Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems

We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass” and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.     

Fully Polynomial Approximation Schemes for a Symmetric Quadratic Knapsack Problem and its Scheduling Applications

We design a fully polynomial-time approximation scheme (FPTAS) for a knapsack problem to minimize a symmetric quadratic function. We demonstrate how the designed FPTAS can be adopted for several single machine scheduling problems to minimize the sum of the weighted completion times. The applications presented in this paper include problems with a single machine non-availability interval (for both the non-resumable and the resumable scenarios) and a problem of planning a single machine maintenance period; the latter problem is closely related to a single machine scheduling problem with two competing agents. The running time of each presented FPTAS is strongly polynomial.     

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q₁–P₀ quadrilateral element and the P₁–P₀ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H²) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|^1/2H³). The methods we study provide an approximate solution (u^h,p^h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.     

An Adaptive P 1 Finite Element Method for Two-Dimensional Maxwell’s Equations

Recently a new numerical approach for two-dimensional Maxwell’s equations based on the Hodge decomposition for divergence-free vector fields was introduced by Brenner et al. In this paper we present an adaptive P ₁ finite element method for two-dimensional Maxwell’s equations that is based on this new approach. The reliability and efficiency of a posteriori error estimators based on the residual and the dual weighted-residual are verified numerically. The performance of the new approach is shown to be competitive with the lowest order edge element of Nédélec’s first family.     

Superconvergence and Extrapolation Analysis of a Nonconforming Mixed Finite Element Approximation for Time-Harmonic Maxwell’s Equations

In this paper, a nonconforming mixed finite element approximating to the three-dimensional time-harmonic Maxwell’s equations is presented. On a uniform rectangular prism mesh, superclose property is achieved for electric field E and magnetic filed H with the boundary condition E×n=0 by means of the asymptotic expansion. Applying postprocessing operators, a superconvergence result is stated for the discretization error of the postprocessed discrete solution to the solution itself. To our best knowledge, this is the first global superconvergence analysis of nonconforming mixed finite elements for the Maxwell’s equations. Furthermore, the approximation accuracy will be improved by extrapolation method.     

On the Optimal Control Problem of a Nonlinear Elliptic Population System

The optimal control problem for a nonlinear elliptic population system is considered. First, under certain hypotheses, the existence and uniqueness of coexistence state solutions are shown. Then the existence of the optimal control is given and the optimality system is established.     

Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ω _j ⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.     

Finite Element Characteristic Methods Requiring no Quadrature

The characteristic methods are known to be very efficient for convection-diffusion problems including the Navier-Stokes equations. Convergence is established when the integrals are evaluated exactly, otherwise there are even cases where divergence has been shown to happen. The family of methods studied here applies Lagrangian convection to the gradients and the function as in Yabe (Comput. Phys. Commun. 66(2–3), 233–242, 1991); the method does not require an explicit knowledge of the equation of the gradients and can be applied whenever the gradients of the convection velocity are known numerically. We show that converge can be second order in space or more. Applications are given for the rotating bell problem.     

Reserve-control Approach to an Optimal Time Control Problem†

The optimal time control problem for single-input sampled data systems is solved using the reserve-control approach. At each sampling instant the time-optimal control is chosen in order to minimize the maximum magnitude of control required at all subsequent sampling intervals. This leads to a simple geometric interpretation regarding the location of certain closed and bounded regions in state space. The resulting optimal control transfers the initial state to the final state in the minimum number of sampling intervals and also provides some reserve control which may be used to counteract random disturbances.     

Convergence Analysis of Primal–Dual Based Methods for Total Variation Minimization with Finite Element Approximation

We consider a minimization model with total variational regularization, which can be reformulated as a saddle-point problem and then be efficiently solved by the primal–dual method. We utilize the consistent finite element method to discretize the saddle-point reformulation; thus possible jumps of the solution can be captured over some adaptive meshes and a generic domain can be easily treated. Our emphasis is analyzing the convergence of a more general primal–dual scheme with a combination factor for the discretized model. We establish the global convergence and derive the worst-case convergence rate measured by the iteration complexity for this general primal–dual scheme. This analysis is new in the finite element context for the minimization model with total variational regularization under discussion. Furthermore, we propose a prediction–correction scheme based on the general primal–dual scheme, which can significantly relax the step size for the discretization in the time direction. Its global convergence and the worst-case convergence rate are also established. Some preliminary numerical results are reported to verify the rationale of considering the general primal–dual scheme and the primal–dual-based prediction–correction scheme.     

Finite Element Meshes Auto-Generation for the Welted Bifurcation

In this paper, firstly, a mathematical model for a specific kind of welted bifurcation is established, the parametric equation for the intersecting curve is resulted in. Secondly, a method for partitioning finite element meshes of the welted bifurcation is put forward, its main idea is that developing the main pipe surface and the branch pipe surface respectively, dividing meshes on each developing plane and obtaining meshes points, then transforming their plane coordinates into space coordinat…     

Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

In this article, a Galerkin finite element approximation for a class of time–space fractional differential equation is studied, under the assumption that \(u_{tt}, u_{ttt}, u_{2\alpha ,tt}\) are continuous for \(\varOmega \times (0,T]\), but discontinuous at time \(t=0\). In spatial direction, the Galerkin finite element method is presented. And in time direction, a Crank–Nicolson time-stepping is used to approximate the fractional differential term, and the product trapezoidal method is employed to treat the temporal fractional integral term. By using the properties of the fractional Ritz projection and the fractional Ritz–Volterra projection, the convergence analyses of semi-discretization scheme and full discretization scheme are derived separately. Due to the lack of smoothness of the exact solution, the numerical accuracy does not achieve second order convergence in time, which is \(O(k^{3-\beta }+k^{3}t_{n+1}^{-\beta }+k^{3}t_{n+1}^{-\beta -1})\), \(n=0,1,\ldots ,N-1\). But the convergence order in time is shown to be greater than one. Numerical examples are also included to demonstrate the effectiveness of the proposed method.     

An Efficient Newton-Krylov Implementation of the Constrained Runs Scheme for Initializing on a Slow Manifold

The long-term dynamic behavior of many dynamical systems evolves on a low-dimensional, attracting, invariant slow manifold, which can be parameterized by only a few variables (“observables”). The explicit derivation of such a slow manifold (and thus, the reduction of the long-term system dynamics) is often extremely difficult or practically impossible. For this class of problems, the equation-free framework has been developed to enable performing coarse-grained computations, based on short full model simulations. Each full model simulation should be initialized so that the full model state is consistent with the values of the observables and close to the slow manifold. To compute such an initial full model state, a class of constrained runs functional iterations was proposed (Gear and Kevrekidis, J. Sci. Comput. 25(1), 17–28, 2005; Gear et al., SIAM J. Appl. Dyn. Syst. 4(3), 711–732, 2005). The schemes in this class only use the full model simulator and converge, under certain conditions, to an approximation of the desired state on the slow manifold. In this article, we develop an implementation of the constrained runs scheme that is based on a (preconditioned) Newton-Krylov method rather than on a simple functional iteration. The functional iteration and the Newton-Krylov method are compared in detail using a lattice Boltzmann model for one-dimensional reaction-diffusion as the full model simulator. Depending on the parameters of the lattice Boltzmann model, the functional iteration may converge slowly or even diverge. We show that both issues are largely resolved by using the Newton-Krylov method, especially when a coarse grid correction preconditioner is incorporated.     

Approximation Algorithms for Scheduling with Reservations

We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations, where m is constant. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan C _max, which is the maximum completion time.     

Multi-step Predictive Control of a PDF-shaping Problem

A predictive control strategy is proposed for the shaping of the output probability density function (PDF) of linear stochastic systems. The B-spline neural network is used to set up the output PDF model and therefore converts the PDF-shaping into the control of B-spline weights vector. The Diophantine equation is then introduced to formulate the predictive PDF model, based on which a moving-horizon control algorithm is developed so as to realize the predictive PDF tracking performance.     

Multigrid Methods for a Mixed Finite Element Method of the Darcy–Forchheimer Model

An efficient nonlinear multigrid method for a mixed finite element method of the Darcy–Forchheimer model is constructed in this paper. A Peaceman–Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.