In this paper, we study the strong stability preserving (SSP) property of a class of deferred correction time discretization methods, for solving the method-of-lines schemes approximating hyperbolic partial differential equations.

For the numerical treatment of Hamiltonian differential equations, symplectic integrators are the most suitable choice, and methods that are conjugate to a symplectic integrator share the same good long-time behavior. This note characterizes linear multistep methods whose underlying one-step method is conjugate to a symplectic integrator. The boundedness of parasitic solution components is not addressed.

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Bénard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in $L^\infty$-norm and optimal error estimates in $L^2$-norm.

This paper is devoted to the study of a nonlinear evolution eddy current model of the type $\partial_t \boldsymbol{B}(\boldsymbol{H})+∇ × (∇ × \boldsymbol{H}) = 0$ subject to homogeneous Dirichlet boundary conditions $\boldsymbol{H} × ν=0$ and a given initial datum. Here, the magnetic properties of a soft ferromagnet are linked by a nonlinear material law described by $\boldsymbol{B}(\boldsymbol{H})$. We apply the backward Euler method for the time discretization and we derive the error estimates in suitable function spaces. The results depend on the nonlinearity of $\boldsymbol{B}(\boldsymbol{H})$.

In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems (steady and nonsteady state) is constructed and analyzed by using linear triangle elements, and the $O(h^2)$-order optimal error estimates are derived under the assumption of strongly regular triangulation and the $H^3$-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.

In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in $\mathbb{R}^d$ ($d$=2,3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.

This paper proposes a reliable and efficient a posteriori error estimator for the finite element methods for the beam problem. It is proved that the error can be bounded by the computable error estimator from above and below up to multiplicative constants that do neither depend on the mesh size nor on the thickness of the beam.

In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, which allows us to obtain approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constant, we can get the exact solution with only one point in each subdomain. Our finite-point method has uniformly convergent rate with respect to wave number $k$ in $L^2$-norm.

The main aim of this paper is to study the convergence properties of a low order mixed finite element for the Stokes problem under anisotropic meshes. We discuss the anisotropic convergence and superconvergence independent of the aspect ratio. Without the shape regularity assumption and inverse assumption on the meshes, the optimal error estimates and natural superconvergence at central points are obtained. The global superconvergence for the gradient of the velocity and the pressure is derived with the aid of a suitable postprocessing method. Furthermore, we develop a simple method to obtain the superclose properties which improves the results of the previous works.

We study in this paper the continuity of the objective function for variable programming. In particular, we study the second-order optimality conditions for unconstrained and constrained variable programming. Some new second-order sufficient and necessary conditions are obtained.