We introduce a low order finite element method for three dimensional elasticity problems. We extend Kouhia-Stenberg element [12] by using two nonconforming components and one conforming component, adding stabilizing terms to the associated bilinear form to ensure the discrete Korn's inequality. Using the second Strang's lemma, we show that our scheme has optimal convergence rates in $L^2$ and piecewise $H^1$-norms even when Poisson ratio $\nu$ approaches $1/2$. Even though some efforts have been made to design a low order method for three dimensional problems in [11,16], their method uses some higher degree basis functions. Our scheme is the first true low order method. We provide three numerical examples which support our analysis. We compute two examples having analytic solutions. We observe the optimal $L^2$ and $H^1$ errors for many different choices of Poisson ratios including the nearly incompressible cases. In the last example, we simulate the driven cavity problem. Our scheme shows non-locking phenomena for the driven cavity problems also.

This work is concerned with numerical schemes for stochastic optimal control problems (SOCPs) by means of forward backward stochastic differential equations (FBSDEs). We first convert the stochastic optimal control problem into an equivalent stochastic optimality system of FBSDEs. Then we design an efficient second order FBSDE solver and an quasi-Newton type optimization solver for the resulting system. It is noticed that our approach admits the second order rate of convergence even when the state equation is approximated by the Euler scheme. Several numerical examples are presented to illustrate the effectiveness and the accuracy of the proposed numerical schemes.

We describe an algorithm that localizes the zeros of a given real $C^2$-function $f$ on an interval $[a,b]$. The algorithm generates a sequence of subintervals which contain a single zero of $f$. In particular, the exact number of zeros of $f$ on $[a,b]$ can be determined in this way. Apart from $f$, the only additional input of the algorithm is an upper and a lower bound for $f''$. We also show how the intervals determined by the algorithm can be further refined until they are contained in the basin of attraction of the Newton method for the corresponding zero.

In this paper, we study the Crank-Nicolson Galerkin finite element method and construct a two-grid algorithm for the general two-dimensional time-dependent Schrödinger equation. Firstly, we analyze the superconvergence error estimate of the finite element solution in $H^1$ norm by use of the elliptic projection operator. Secondly, we propose a fully discrete two-grid finite element algorithm with Crank-Nicolson scheme in time. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two Poisson equations on the fine grid. Finally, we also derive error estimates of the two-grid finite element solution with the exact solution in $H^1$ norm. It is shown that the solution of two-grid algorithm can achieve asymptotically optimal accuracy as long as mesh sizes satisfy $H = \mathcal{O}(h^{\frac{1}{2}})$.

In this paper, we consider the unilateral obstacle problem, trying to find the numerical solution and coincidence set. We construct an equivalent format of the original problem and  propose a method with a second-order in time dissipative system for solving the equivalent format. Several numerical examples are given to illustrate the effectiveness and stability of the proposed algorithm. Convergence speed comparisons with existent numerical algorithm are also provided and our algorithm is fast.

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e.,  $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.

In the present work, linear combinations of Caputo fractional derivatives are fast evaluated based on the efficient sum-of-exponentials (SOE) approximation for  kernels in Caputo fractional derivatives with an absolute error $\epsilon,$  which is a further work of the existing results in [13]  (Commun. Comput. Phys., 21 (2017), pp. 650-678) and [16] (Commun. Comput. Phys., 22 (2017), pp. 1028-1048). Both the storage needs and computational amount are  significantly reduced compared with the direct algorithm. Applications of the proposed fast algorithm are illustrated by solving a second-order multi-term time-fractional sub-diffusion problem. The unconditional stability and convergence of the fast difference scheme are proved. The CPU time is largely reduced while the accuracy is kept, especially for the cases of  large  temporal level, which is displayed by  numerical experiments.

Based on the generalized Laguerre and Hermite functions, we construct two types of Birkhoff-type interpolation basis functions. The explicit expressions are derived, and fast and stable algorithms are provided for computing these basis functions. As applications, some well-conditioned collocation methods are proposed for solving various second-order differential equations in unbounded domains. Numerical experiments illustrate that our collocation methods are more efficient than the standard Laguerre/Hermite collocation approaches.

A Godunov-type finite volume scheme on unstructured grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate. We show the direct observation that the model isn't a Galilean invariant system. At the cell boundary, the modified Harten-Lax-van Leer (HLL) approximate Riemann solver is adopted to calculate the numerical flux. The modified HLL flux is not troubled by the lack of Galilean invariance of the model and it is helpful to handle discontinuities at free interface. Rigidly the system is not always a hyperbolic system due to the dependence of flux on the velocity gradient. Even so, our numerical results still show quite good agreements with reference solutions. The simulations for granular avalanche flows with shock waves indicate that the scheme is applicable.

This paper aims to develop high-order numerical methods for solving the system partial differential equations (PDEs) and partial integro-differential equations (PIDEs) arising in exotic option pricing under regime-switching models and regime-switching jump-diffusion models, respectively. Using cubic Hermite polynomials, the high-order collocation methods are proposed to solve the system PDEs and PIDEs. This collocation scheme has the second-order convergence rates in time and fourth-order rates in space. The computation of the Greeks for the options is also studied. Numerical examples are carried out to verify the high-order convergence and show the efficiency for computing the Greeks.

In this paper, we first present an adaptive nonmonotone term to improve the efficiency of nonmonotone line search, and then an active set identification technique is suggested to get more efficient descent direction such that it improves the local convergence behavior of algorithm and decreases the computation cost. By means of the adaptive nonmonotone line search and the active set identification technique, we put forward a global convergent gradient-based method to solve the nonnegative matrix factorization (NMF) based on the alternating nonnegative least squares framework, in which we introduce a modified Barzilai-Borwein (BB) step size. The new modified BB step size and the larger step size strategy are exploited to accelerate convergence. Finally, the results of extensive numerical experiments using both synthetic and image datasets show that our proposed method is efficient in terms of computational speed.

In this article, we explore the numerical approximation for a phase-field model to the moving contact line (MCL) problem, governed by the Cahn-Hilliard equation with a dynamic contact angle condition. A difficult and challenging issue comes from the high order, nonlinear and coupled time marching system with a small dimensionless parameter describing the interface thickness. The scalar auxiliary variable (SAV) approach has been recently developed for a large class of gradient flows in principle. Due to the nonlinear gradient terms in both the bulk and boundary energies, we introduce multiple scalar auxiliary variables (MSAV) approach rather than single SAV scheme to deal with phase field contact line model. The MSAV scheme numerically gives a better description of the contact line dynamic  in terms of the difference between the modified and original energies. Moreover, the resulting numerical schemes enjoy the advantages of second order accuracy, unconditional energy stability, and only require solving a linear decoupled system with constant coefficients at each time step. Numerical experiments are presented to verify the effectiveness and efficiency of the proposed schemes for a wide range of mobility parameters and phenomenological boundary relaxation parameters.