A new linear and conservative finite difference scheme which preserves discrete mass and energy is developed for the two-dimensional Gross–Pitaevskii equation with angular momentum rotation. In addition to the energy estimate method and mathematical induction, we use the lifting technique as well as some well-known inequalities to establish the optimal $H^{1}$-error estimate for the proposed scheme with no restrictions on the grid ratio. Unlike the existing numerical solutions which are of second-order accuracy at the most, the convergence rate of the numerical solution is proved to be of order $O(h^{4}+\unicode[STIX]{x1D70F}^{2})$ with time step $\unicode[STIX]{x1D70F}$ and mesh size $h$. Numerical experiments have been carried out to show the efficiency and accuracy of our new method.

This paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

The semilocal convergence of a third-order Newton-like method for solving nonlinear equations is considered. Under a weak condition (the so-called γ-condition) on the derivative of the nonlinear operator, we establish a new semilocal convergence theorem for the Newton-like method and also provide an error estimate. Some numerical examples show the applicability and efficiency of our result, in comparison to other semilocal convergence theorems.

In error estimates of various numerical approaches for solving decoupled forward backward stochastic differential equations (FBSDEs), the rate of convergence for one variable is usually less than for the other. Under slightly strengthened smoothness assumptions, we show that the fully discrete Euler scheme admits a first-order rate of convergence for both variables.

Pressure-correction projection finite element methods (FEMs) are proposed to solve nonstationary natural convection problems in this paper. The first-order and second-order backward difference formulas are applied for time derivative, the stability analysis and error estimates of the semi-discrete schemes are presented using energy method. Compared with characteristic variational multiscale FEM, pressure-correction projection FEMs are more efficient and unconditionally energy stable. Ample numerical results are presented to demonstrate the effectiveness of the pressure-correction projection FEMs for solving these problems.

In this paper, a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection – diffusion problem is analyzed. The method is shown to be convergent uniformly in the perturbation parameter ∈ provided only that ∈ ≤ N–1. An convergent rate in a discrete streamline-diffusion norm is established under certain regularity assumptions. Finally, through numerical experiments, we verified the theoretical results.

In this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.

This paper analyzes semi-discrete and fully discrete hybrid stress quadrilateral finite element methods for 2-dimensional linear elastodynamic problems. The methods use a 4 node hybrid stress quadrilateral element in the space discretization. In the fully discrete scheme, an implicit second-order scheme is adopted in the time discretization. We derive optimal a priori error estimates for the two schemes and an unconditional stability result for the fully discrete scheme. Numerical experiments confirm the theoretical results.

Upon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

In this paper, we introduce and study a new method for solving inverse source problems, through a working model that arises in bioluminescence tomography (BLT). In the BLT problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter for trade off between the accuracy and stability of approximate solutions. The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition, followed by the Tikhonov regularization. By properly adjusting the parameter in the Robin boundary condition, we achieve two important properties for our new method: first, the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy; second, the convergence order of the regularized solutions reaches one with respect to the noise level. Then, the finite element method is used to compute numerical solutions and a new finite element error estimate is derived for discrete solutions. These results improve related results found in the existing literature. Several numerical examples are provided to illustrate the theoretical results.

An inverse diffraction problem is considered. Both classical Tikhonov regularisation and a slow-evolution-from-the-continuation-boundary (SECB) method are used to solve the ill-posed problem. Regularisation error estimates for the two methods are compared, and the SECB method is seen to be an improvement on the classical Tikhonov method. Two numerical examples demonstrate their feasibility and efficiency.

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete l2 norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

This paper studies a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

An explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

A semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.

In this work, we are concerned with a time-splitting Fourier pseudospectral (TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless parameter ɛ ∊ (0,1]. In the nonrelativistic limit regime, the small ɛ produces high oscillations in exact solutions with wavelength of (ɛ2) in time. The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time, with both the nonlinear and linear subproblems exactly integrable in time and, respectively, Fourier frequency spaces. The method is fully explicit and time reversible. Moreover, we establish rigorously the optimal error bounds of a second-order TSFP for fixed ɛ = (1), thanks to an observation that the scheme coincides with a type of trigonometric integrator. As the second task, numerical studies are carried out, with special efforts made to applying the TSFP in the nonrelativistic limit regime, which are geared towards understanding its temporal resolution capacity and meshing strategy for (ɛ2)-oscillatory solutions when 0 < ɛ « 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic (ɛ−2Δt2) temporal discretization error bound (Δt refers to time step). On the other hand, the temporal error bounds for most trigonometric integrators, such as the well-established Gautschi-type integrator in, are (ɛ−4Δt2). Thus, our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime. These results, either rigorous or numerical, are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.

In this paper, we revisit a priori error analysis of nonconforming finite element methods for the Poisson problem. Based on some techniques developed in the context of the a posteriori error analysis, under two reasonable assumptions on the nonconforming finite element spaces, we prove that, up to some oscillation terms, the consistency error can be bounded by the approximation error. We check these two assumptions for the most used lower order nonconforming finite element methods. Compared with the classical error analysis of the nonconforming finite element method, the a priori analysis herein only needs the H1 regularity of the exact solution.

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The contribution of the paper is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem; 3) some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition, and that the convergence rate of the method is exponential if the optimal initial condition is smooth.

In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is L2 stable. When the finite element space consists of interpolative polynomials of degrees k, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of . Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size H, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size . These methods we studied provide an approximate solution with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.