In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method.

This is one of our series works on numerical methods for mean-field forward backward stochastic differential equations (MFBSDEs). In this work, we propose an explicit multistep scheme for MFBSDEs which is easy to implement, and is of high order rate of convergence. Rigorous error estimates of the proposed multistep scheme are presented. Numerical experiments are carried out to show the efficiency and accuracy of the proposed scheme.

In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order $1,2,3,\ldots$ Mean-square stability  has been investigated for full implicit strong Stratonovich-Taylor scheme with order $2$, and it has larger mean-square stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order $2$. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift  and the diffusion terms. Our numerical experiment shows these points.

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.

In this paper, we consider the knot placement problem in B-spline curve approximation. A novel two-stage framework is proposed for addressing this problem. In the first step, the $l_{\infty, 1}$-norm model is introduced for the sparse selection of candidate knots from an initial knot vector. By this step, the knot number is determined. In the second step, knot positions are formulated into a nonlinear optimization problem and optimized by a global optimization algorithm — the differential evolution algorithm (DE). The candidate knots selected in the first step are served for initial values of the DE algorithm. Since the candidate knots provide a good guess of knot positions, the DE algorithm can quickly converge. One advantage of the proposed algorithm is that  the knot number and knot positions are determined automatically. Compared with the current existing algorithms, the proposed algorithm finds approximations with smaller fitting error when the knot number is fixed in advance. Furthermore, the proposed algorithm is robust to noisy data and can handle with few data points. We illustrate with some examples and applications.

In this paper, we consider the Euler-Maruyama method for a class of stochastic Volterra integral equations (SVIEs). It is known that the strong convergence order of the Euler-Maruyama method is $\frac12$. However, the strong superconvergence order $1$ can be obtained for a class of SVIEs if the kernels $\sigma_{i}(t, t) = 0$ for $i=1$ and $2$; otherwise, the strong convergence order is $\frac12$. Moreover, the theoretical results are illustrated by some numerical examples.

In this paper, we construct an $H^1$-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates. The element has optimal approximation rates. Using this quadratic element, two stable discretizations for the Stokes equations are developed, which can be viewed as the extensions of the $P_2$-$P_0$ and the $Q_2$-(discontinuous)$P_1$ elements, respectively, to polygonal meshes. Numerical results are presented, which support our theoretical claims.

In this paper, we derive and analyse waveform relaxation (WR) methods for solving differential equations evolving on a Lie-group. We present both continuous-time and discrete-time WR methods and study their convergence properties. In the discrete-time case, the novel methods are constructed by combining WR methods with Runge-Kutta-Munthe-Kaas (RK-MK) methods. The obtained methods have both advantages of WR methods and RK-MK methods, which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a manifold. Three numerical experiments are given to illustrate the feasibility of the new WR methods.