In this work, we propose a simple yet effective gradient projection algorithm for a class of stochastic optimal control problems. The basic iteration block is to compute gradient projection of the objective functional by solving the state and co-state equations via some Euler methods and by using the Monte Carlo simulations. Convergence properties are discussed and extensive numerical tests are carried out. Possibility of extending this algorithm to more general stochastic optimal control is also discussed.

In this paper, we develop a new energy-conserved S-FDTD scheme for the Maxwell's equations in metamaterials. We first derive out the new property of energy conservation of the governing equations in metamaterials, and then propose the energy-conserved S-FDTD scheme for solving the problems based on the staggered grids. We prove that the proposed scheme is energy-conserved in the discrete form and unconditionally stable. Based on the energy method, we further prove that the scheme for the Maxwell's equations in metamaterials is first order in time and second order in space. Numerical experiments are carried out to confirm the energy conservation and the convergence rates of the scheme. Moreover, numerical examples are also taken to show the propagation features of electromagnetic waves in the DNG metamaterials.

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by $\varepsilon^2$, where $\varepsilon$ is the perturbation parameter, $\varepsilon\in (0,1]$. For small values of $\varepsilon$, an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition $N^{-1} = o(\varepsilon)$, $N^{-1}_0 = o(1)$, where $N +1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges $\varepsilon$-uniformly at the rate $\mathcal{O}(N^{-1}lnN+N^{-1}_0)$. Numerical experiments confirm the theoretical results.

The non-local diffusion model provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities, which cannot be described properly by classical theory of solid mechanics. However, because the non-local nature of the non-local diffusion operator, the numerical methods for non-local diffusion model generate dense or even full stiffness matrices. A direct solver typically requires $O(N^3)$ of operations and $O(N^2)$ of memory where $N$ is the number of unknowns. We develop a fast collocation method for the non-local diffusion model which has the following features: (i) It reduces the computational cost from $O(N^3)$ to $O(N log^2 N)$ and memory requirement from $O(N^2)$ to $O(N)$. (ii) It requires only one-fold integration in the evaluation of the stiffness matrix. Numerical experiments show the utility of the method.

In this paper, we obtain optimal first order error estimates for a fully discrete fractional-step scheme applied to the Navier-Stokes equations. This scheme uses decomposition of the viscosity in time and finite elements (FE) in space.
In [15], optimal first order error estimates (for velocity and pressure) for the corresponding time-discrete scheme were obtained, using in particular $H^2 \times H^1$ estimates for the approximations of the velocity and pressure. Now, we use this time-discrete scheme as an auxiliary problem to study a fully discrete finite element scheme, obtaining optimal first order approximation for velocity and pressure with respect to the max-norm in time and the $H^1 \times L^2$-norm in space.
The proof of these error estimates are based on three main points: a) provide some new estimates for the time-discrete scheme (not proved in [15]) which must be now used, b) give a discrete version of the $H^2 \times H^1$ estimates in FE spaces, using stability in the $W^{1,6} \times L^6$-norm of the FE Stokes projector, and c) the use of a weight function vanishing at initial time will let to hold the error estimates without imposing global compatibility for the exact solution.

In this paper, we introduce a coupled approach of local discontinuous Galerkin (LDG) and continuous finite element method (CFEM) for solving singularly perturbed convection-diffusion problems. When the coupled continuous-discontinuous linear FEM is used under the Shishkin mesh, a uniform convergence rate $O(N^{-1}ln N)$ in an associated norm is established, where $N$ is the number of elements. Numerical experiments complement the theoretical results. Moreover, a uniform convergence rate $O(N^{-2})$ in $L^2$ norm, is observed numerically on the Shishkin mesh.

We derive error estimates in the $L_2$ norms, for the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element methods for steady state, energy dependent, Fermi equation in three space dimensions. These estimates yield optimal convergence rates due to the maximal available regularity of the exact solution. Here our focus is on theoretical aspects of the $h$ and $hp$ approximations in both SD and DG settings.

In this paper, we study error estimates of a special $\theta$-scheme — the Crank-Nicolson scheme proposed in [25] for solving the backward stochastic differential equation with a general generator, $-dy_t = f(t, y_t, z_t)dt-z_tdW_t$. We rigorously prove that under some reasonable regularity conditions on $\varphi$ and $f$, this scheme is second-order accurate for solving both $y_t$ and $z_t$ when the errors are measured in the $L^p (p \geq 1)$ norm.

Least-squares(LS) finite element methods are applied successfully to a wide range of problems arising from science and engineering. However, there are reservations to use LS methods for problems with low regularity solutions. In this paper, we consider LS methods for second-order elliptic problems using the minimum regularity assumption, i.e. the solution only belongs to $H^1$ space. We provide a theoretical analysis showing that LS methods are competitive alternatives to mixed and standard Galerkin methods by establishing that LS solutions are bounded by the mixed and standard Galerkin solutions.

The main aim of this paper is to study the approximation of nonconforming mixed finite element methods for stationary, incompressible magnetohydrodynamics (MHD) equations in 3D. A family of nonconforming finite elements are taken as the approximation spaces for the velocity field, the piecewise constant element for the pressure and the Nédélec's element with the lowest order for the magnetic field on hexahedra or tetrahedra. A new simple method is adopted to prove the discrete Poincaré-Friedrichs inequality instead of the discrete Helmholtz decomposition method. The existence and uniqueness of the approximate solutions are shown. The convergence analysis is presented and the optimal order error estimates for the pressure in $L^2$-norm, as well as those in a broken $H^1$-norm for the velocity field and H($curl$)-norm for the magnetic field are derived.

Under certain kinetic or thermodynamic conditions, proteins make large conformational changes, formally called state transitions, resulting in significant changes in their chemical or biological functions. These dynamic properties of proteins can be studied through molecular dynamics simulation. However, in contrast to conventional dynamics simulation protocols where an initial-value problem is solved, the simulation of transition of protein conformation can be done by solving a boundary-value problem, with the beginning and ending states of the protein as the boundary conditions. While a boundary-value problem is generally more difficult to solve, it provides a more realistic model for transition of protein conformation and has certain computational advantages as well, especially for long-time simulations. Here we study the solution of the boundary-value problems for the simulation of transition of protein conformation using a standard class of numerical methods called the multiple shooting methods. We describe the methods and discuss the issues related to their implementations for our specific applications, including the definition of the boundary conditions, the formation of the initial trajectories, and the convergence of the solutions. We present the results from using the multiple shooting methods for the study of the conformational transition of a small molecular cluster and an alanine dipeptide, and show the potential extension of the methods to larger biomolecular systems.

This paper presents the stabilities for both two modular, projection-based variational multiscale (VMS) methods and the error analysis for only first one for the incompressible Navier-Stokes equations, extending the analysis in [39] to include nonlinear eddy viscosities. In VMS methods, the influence of the unresolved scales onto the resolved small scales is modeled by a Smagorinsky-type turbulent viscosity acting only on the marginally resolved scales. Different realization of VMS models arise through different models of fluctuations. We analyze a method of inducing a VMS treatment of turbulence in an existing NSE discretization through an additional, uncoupled projection step. We prove stability, identifying the VMS model and numerical dissipation and give an error estimate. Numerical tests are given that confirm and illustrate the theoretical estimates. One method uses a fully nonlinear step inducing the VMS discretization. The second induces a nonlinear eddy viscosity model with a linear solve of much less cost.

The optimal $\mathcal{H}_2$ model reduction is an important tool in studying dynamical systems of a large order and their numerical simulation. We formulate the reduction problem as a minimization problem over the Grassmann manifold. This allows us to develop a fast gradient flow algorithm suitable for large-scale optimal $\mathcal{H}_2$ model reduction problems. The proposed algorithm converges globally and the resulting reduced system preserves stability of the original system. Furthermore, based on the fast gradient flow algorithm, we propose a sequentially quadratic approximation algorithm which converges faster and guarantees the global convergence. Numerical examples are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed algorithms.