Generalized Bézier surfaces are a multi-sided generalization of classical tensor product Bézier surfaces with a simple control structure and inherit most of the appealing properties from Bézier surfaces. However, the original degree elevation changes the geometry of generalized Bézier surfaces such that it is undesirable in many applications, e.g. isogeometric analysis. In this paper, we propose an improved degree elevation algorithm for generalized Bézier surfaces preserving not only geometric consistency but also parametric consistency. Based on the knot insertion of B-splines, a novel knot insertion algorithm for generalized Bézier surfaces is also proposed. Then the proposed algorithms are employed to increase degrees of freedom for multi-sided computational domains parameterized by generalized Bézier surfaces in isogeometric analysis, corresponding to the traditional $p$-, $h$-, and $k$-refinements. Numerical examples demonstrate the effectiveness and superiority of our method.

For solving the stochastic differential equations driven by fractional Brownian motion, we present the modified split-step theta method by combining truncated Euler-Maruyama method with split-step theta method. For the problem under a locally Lipschitz condition and a linear growth condition, we analyze the strong convergence and the exponential stability of the proposed method. Moreover, for the stochastic delay differential equations with locally Lipschitz drift condition and globally Lipschitz diffusion condition, we give the order of convergence. Finally, numerical experiments are done to confirm the theoretical conclusions.

In this paper, we present a novel penalty model called ExPen for optimization over the Stiefel manifold. Different from existing penalty functions for orthogonality constraints, ExPen adopts a smooth penalty function without using any first-order derivative of the objective function. We show that all the first-order stationary points of ExPen with a sufficiently large penalty parameter are either feasible, namely, are the first-order stationary points of the original optimization problem, or far from the Stiefel manifold. Besides, the original problem and ExPen share the same second-order stationary points. Remarkably, the exact gradient and Hessian of ExPen are easy to compute. As a consequence, abundant algorithm resources in unconstrained optimization can be applied straightforwardly to solve ExPen.

We propose a simple iterative (SI) algorithm for the maxcut problem through fully using an equivalent continuous formulation. It does not need rounding at all and has advantages that all subproblems have explicit analytic solutions, the cut values are monotonically updated and the iteration points converge to a local optima in finite steps via an appropriate subgradient selection. Numerical experiments on G-set demonstrate the performance. In particular, the ratios between the best cut values achieved by SI and those by some advanced combinatorial algorithms in [Ann. Oper. Res., 248 (2017), 365–403] are at least 0.986 and can be further improved to at least 0.997 by a preliminary attempt to break out of local optima.

In this paper, we present a nonlinear correction technique to modify the nine-point scheme proposed in [SIAM J. Sci. Comput., 30:3 (2008), 1341-1361] such that the resulted scheme preserves the positivity. We first express the flux by the cell-centered unknowns and edge unknowns based on the stencil of the nine-point scheme. Then, we use a nonlinear combination technique to get a monotone scheme. In order to obtain a cell-centered finite volume scheme, we need to use the cell-centered unknowns to locally approximate the auxiliary unknowns. We present a new method to approximate the auxiliary unknowns by using the idea of an improved multi-points flux approximation. The numerical results show that the new proposed scheme is robust, can handle some distorted grids that some existing finite volume schemes could not handle, and has higher numerical accuracy than some existing positivity-preserving finite volume schemes.

In this paper two dimensional elliptic interface problem with imperfect contact is considered, which is featured by the implicit jump condition imposed on the imperfect contact interface, and the jumping quantity of the unknown is related to the flux across the interface. A finite difference method is constructed for the 2D elliptic interface problems with straight and curve interface shapes. Then, the stability and convergence analysis are given for the constructed scheme. Further, in particular case, it is proved to be monotone. Numerical examples for elliptic interface problems with straight and curve interface shapes are tested to verify the performance of the scheme. The numerical results demonstrate that it obtains approximately second-order accuracy for elliptic interface equations with implicit jump condition.

In this paper, we propose a numerical method for turning point problems in one dimension based on Petrov-Galerkin finite element method (PGFEM). We first give a priori estimate for the turning point problem with a single boundary turning point. Then we use PGFEM to solve it, where test functions are the solutions to piecewise approximate dual problems. We prove that our method has a first-order convergence rate in both $L^∞_h$ norm and a discrete energy norm when we select the exact solutions to dual problems as test functions. Numerical results show that our scheme is efficient for turning point problems with different types of singularities, and the convergency coincides with our theoretical results.

In this work, we analyze the three-step backward differentiation formula (BDF3) method for solving the Allen-Cahn equation on variable grids. For BDF2 method, the discrete orthogonal convolution (DOC) kernels are positive, the stability and convergence analysis are well established in [Liao and Zhang, Math. Comp., 90 (2021), 1207–1226] and [Chen, Yu, and Zhang, arXiv:2108.02910, 2021]. However, the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels. By developing a novel spectral norm inequality, the unconditional stability and convergence are rigorously proved under the updated step ratio restriction $r_k :=\tau_k/\tau_{k−1}≤1.405$ for BDF3 method. Finally, numerical experiments are performed to illustrate the theoretical results. To the best of our knowledge, this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.

We consider a numerical algorithm for the two-dimensional time-harmonic elastic wave scattering by unbounded rough surfaces with Dirichlet boundary condition. A Nyström method is proposed for the scattering problem based on the integral equation method. Convergence of the Nyström method is established with convergence rate depending on the smoothness of the rough surfaces. In doing so, a crucial role is played by analyzing the singularities of the kernels of the relevant boundary integral operators. Numerical experiments are presented to demonstrate the effectiveness of the method.