In this paper, we design novel high order probabilistic numerical algorithms for forward backward stochastic differential equations. Moreover, we derive the error estimates and prove the high order convergence rates of the proposed schemes. Because the proposed scheme involves conditional expectations, an estimator based on the multilevel Monte Carlo method is applied to approximate the conditional expectations. Furthermore, we theoretically demonstrate that the computational complexity of our numerical method is proportional to the square of prescribed accuracy. Numerical experiments are given to illustrate the theoretical results.

This paper is devoted to identifying the source term and initial value simultaneously in a time-fractional Black-Scholes equation, which is an ill-posed problem. The inverse problem is transformed into a system of operator equations, and under certain source conditions, conditional stability is established. We propose a regularization method with two differential operators to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived, respectively. Numerical experiments are presented to validate the effectiveness of the proposed regularization method.

Superconvergent behavior for nonlinear Kirchhoff-type with damping is researched by a structure-preserving nonconforming finite element method (FEM). A new implicit energy dissipation scheme is developed and the numerical solution is bounded in energy norm. The existence of the numerical solution is obtained with the help of the Brouwer fixed-point theorem and then the uniqueness is gained. Superconvergence characteristics is revealed by the properties of the nonconforming FE and a special splitting technique. Numerical tests confirm the correctness of the theoretical research results.

In this paper, we analyze the virtual element method for the symmetric second order elliptic eigenvalue problem with variable coefficients in two and three dimensions, which reduces the number of degrees of freedom of the standard virtual element method. We attempt to prove the interpolation theory and stability analysis for the serendipity nodal virtual element space, which provides new stabilization terms for the virtual element schemes. Then we prove the spectral approximation and the optimal a priori error estimates. Moreover, we construct a fully computable residual-type a posteriori error estimator applied to the adaptive serendipity virtual element method and prove its upper and lower bounds with respect to the approximation error. Finally, we show numerical examples to verify the theoretical results and show the comparison between standard and serendipity virtual element methods.

Three multi-level mixed finite element methods for the steady Boussinesq equations are analyzed and discussed in this paper. The nonlinear and multi-variables coupled problem on a coarse mesh with the mesh size $h_0$ is solved firstly, and then, a series of decoupled and linear subproblems with the Stokes, Oseen and Newton iterations are solved on the successive and refined grids with the mesh sizes $h_j$, $j$ = 1, 2, . . . , $J$. The computational scales are reduced and the computational costs are saved. Furthermore, the uniform stability and convergence results in both $L^2$ - and $H^1$ -norms of are derived under some uniqueness conditions by using the mathematical induction and constructing the dual problems. Theoretical results show that the multi-level methods have the same order of numerical solutions in the $H^1$ -norm as the one level method with the mesh sizes $h_j$ = $h^2_j$−1, $j$ = 1, 2, . . . , $J$. Finally, some numerical results are provided to investigate and compare the effectiveness of the multi-level mixed finite element methods.

This paper develops a low order weak Galerkin (WG) finite element method for the steady thermally coupled incompressible magnetohydrodynamics flow. In the interior of elements, the WG scheme uses piecewise linear polynomials for the approximations of the velocity, the magnetic field and the temperature, and piecewise constants for the approximations of the pressure and the magnetic pseudo-pressure; and on the interfaces of elements, the scheme uses piecewise constants for the numerical traces of velocity and the temperature, and piecewise linear polynomials for the numerical traces of the magnetic fields, the pressure and the magnetic pseudo-pressure. This WG method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results as well as optimal a priori error estimates for the discrete scheme are obtained. A convergent linearized iterative algorithm is presented. Numerical experiments are provided to verify the theoretical analysis.

In this paper, two $\mathcal{O}$($h^2$)-accurate conservative finite element schemes with low-degree polynomials for the incompressible Stokes equations are presented. The schemes use respective $H$(div) finite element spaces, namely the third-order Brezzi-Douglas-Marini space and Brezzi-Douglas-Fortin-Marini space, with enhanced smoothness for the velocity and piecewise quadratic polynomials for the pressure, and are denoted as $sBDM_3$ − $P_2$ and $sBDFM_3$ − $P_2$ schemes, respectively. The discrete Korn inequality holds for both $sBDM_3$ and $sBDFM_3$ finite element spaces. For the $sBDM_3$ − $P_2$ scheme, the inf-sup condition holds on general triangulations, and for the $sBDFM_3$ − $P_2$ scheme, the inf-sup condition holds on triangulations with mild restriction. Both schemes achieve an energy norm of velocity errors of $\mathcal{O}$($h^2$) order and an $L^2$ -norm of pressure errors of $\mathcal{O}$($h^2$) order. Numerical experiments support the theoretical constructions.

An improved nonlinear fourth-order Cahn-Hilliard (INFOCH) equation is first developed to ensure that its numerical model is symmetric, positive definite, and solvable. Then, by introducing an auxiliary function, the INFOCH equation is decomposed into the nonlinear system of equations with second-order derivatives of spatial variables. Subsequently, by using the Crank-Nicolson (CN) technique to discretize the time derivative, a new time semi-discretized mixed CN (TSDMCN) scheme with second-order accuracy is constructed, and the existence, stability, and error estimates of TSDMCN solutions are analyzed. Thenceforth, a new two-grid mixed finite element (MFE) CN (TGMFECN) method is created by using two-grid MFE method to discretize the TSDMCN scheme, and the existence, stability, and error estimates of TGMFECN solutions are discussed. Next, it is most important that by using proper orthogonal decomposition to reduce the dimension of unknown coefficient vectors of TGMFECN solutions and keep the MFE basis functions unchanged, a new TGMFECN dimensionality reduction (TGMFECNDR) method with very few unknowns, unconditional stability, and second-order time precision is created, and the existence, stability, and error estimates of TGMFECNDR solutions are proved. Finally, the superiority of TGMFECNDR method and the correctness of the obtained theoretical results are showed by two sets of numerical experiments.

In this paper, a numerical method for solving nonlinear stochastic delay differential equations is proposed: two-step Milstein method. The mean square consistent and mean-square convergence of the numerical method are studied. Through the relevant derivation, the conditions that the coefficients need to be satisfied when the numerical method is mean-square consistent and mean-square convergent are obtained, and it is proved that the mean-square convergence order of the numerical method is 1. Finally, the theoretical results are verified by numerical experiments.

Inspired by Polyak’s heavy-ball method, this paper proposes an adaptive deterministic block coordinate descent method with momentum (mADBCD) for efficiently solving large-scale linear least-squares problems. The proposed method introduces a novel column selection criterion based on the Euclidean norm of the residual vector of the normal equation. In contrast to classical block coordinate descent methods, mADBCD does not require a fixed pre-partitioning of the column indices of the coefficient matrix and avoids the expensive computation of Moore Penrose pseudoinverses of submatrices at each iteration. The method adaptively updates the block index set at each step, thereby improving both flexibility and scalability. When the coefficient matrix is of full column rank, we prove that mADBCD converges linearly to the unique solution of the least-squares problem. Numerical experiments are conducted to show that mADBCD outperforms several recent block coordinate descent methods in terms of iteration count and CPU time. In particular, when solving extremely sparse least-squares problems, mADBCD is the first block coordinate descent method reported to achieve CPU time nearly comparable to that of the classical least squares QR (LSQR) method [Paige and Saunders, ACM Trans. Math. Softw., 8 (1982)].