This article proposes a new $P_{1}$ nonconforming Nitsche's extended finite element method for elliptic interface problems with interface-unfitted meshes. It is shown that the stability of the discrete formulation is independent of not only the mesh size and the diffusion parameters, but also the position of the interface, showing a robustness over the location of interface. In spite of the low regularity of interface problems, the optimal convergence is obtained. Numerical experiments are carried out to validate theoretical results.

In this paper, an efficient numerical method is proposed for the valuation of American better-of options based on the Black-Scholes model. Because of the existence of the optimal exercise boundary, the American better-of option satisfies a two dimensional parabolic linear complementarity problem on an unbounded domain. We first transform it into a one dimensional free boundary problem by a standard change of variables. And then the front-fixing transformation and the far field truncation are used to deal with the free boundary and the unbounded domain in succession, which yields a parabolic problem with unknown coefficient (free boundary) on a bounded regular domain. Furthermore, a finite element method is applied to discretize the resulting continuous system. The stability of the semi-discrete solution is also established. Meanwhile, Newton's method is used to solve the discretized system to obtain the free boundary and the option value simultaneously. The nonnegativity of the iteration solutions is also proved. Finally, numerical simulations are carried out to test the performance of the proposed method.

To overcome the difficulty for solving fourth order partial differential equations (PDEs) using localized methods, we introduce and extend a recent method to decompose the particular solution of such equation into particular solutions of two second-order differential equations using radial basis functions (RBFs). In this way, the localized method of approximate particular solutions (LMAPS) can be used to directly solve a fourth-order PDE without splitting it into two second-order problems. The closed-form particular solutions for polyharmonic splines RBFs augmented with polynomial basis functions for Helmholtz-type equations are the cores of the solution process. Several novel techniques are proposed to further improve the accuracy and efficiency. Four numerical examples are presented to show the effectiveness of our approach.

In this paper, a localized space-time method of fundamental solutions (LSTMFS) is proposed to solve the diffusion and convection-diffusion problems. The proposed LSTMFS only requires some arbitrarily-distributed nodes inside the space-time domain and along its boundary. The local subdomain corresponding to each node can firstly be determined based on the Euclidean distance between the nodes. Then, the variable at each node can be expressed as a linear combination of variables at its supporting nodes. By solving a resultant sparse system, the variable at any node in the considered space-time domain can be obtained. Compared with the traditional space-time method of fundamental solutions, the proposed LSTMFS is more suitable for solving large-scale and long-time diffusion problems. Furthermore, the LSTMFS without temporal-difference is simple, accurate and easy-to-implement due to its semi-analytical and meshless features. Numerical experiments, including diffusion and convection-diffusion problems, confirm the validity and accuracy of the proposed LSTMFS.

In this paper, we define a new nonconforming finite element space on parallelepiped. Using our new nonconforming space and a vector part of  Kim-Kwak mixed finite element space, we suggest a new class of higher order mixed finite volume method. We show that the mixed finite volume methods can be implemented by solving the primal problem with our new nonconforming finite element methods for the pressure variable. And we can obtain the velocity variable by local recovery technique. An optimal error analysis is given and also numerical results are presented to support our analysis.

High order weighted compact nonlinear scheme (WCNS) has become an alternative method of finite difference weighted essentially non-oscillatory (WENO) scheme in many different research areas due to its better spectral properties. However, its heavy computational time even more expensive than the classical WENO scheme is still a bottleneck problem. To relieve it in a sense, a framework of high order hybrid WCNS (HWCNS) combining the weighted nonlinear interpolations proposed in [Deng et al., JCP, 165] or [Zhang et al., JCP, 227] in the non-smooth stencils with corresponding linear compact interpolations in the smooth stencils respectively is designed for solving the hyperbolic conservation laws in this work. A newly developed high order shock detector based on the radial basis function, which can capture the locations of shocks and high gradients accurately and sharply, is used to measure the smoothness of the solution at each grid point. The HWCNS demonstrates higher resolution, fewer dissipation/dispersion errors, less computational time in the extensive one- and two-dimensional classical examples by comparing with the WCNS.

In this work, a fourth order weighted essentially non-oscillatory (WENO) scheme is developed for hyperbolic conservation laws. The new reconstruction is a convex combination of three linear reconstructions. To keep high order accuracy in smooth regions and maintain non-oscillatory near discontinuities, the nonlinear weights are carefully designed. The main advantage of the proposed scheme is that the scheme achieves one order of improvement in accuracy in smooth regions compared with the classical third order scheme when using the same spatial nodes. Several benchmark examples are presented to verify the scheme's fourth order accuracy and capacity of dealing with problems containing complicated structures.

In this paper, we study the mixed element schemes of the Reissner-Mindlin plate model and the Kirchhoff plate model in multiply-connected domains. Constructing a regular decomposition of $H_0(rot,\Omega)$ and a Helmholtz decomposition of its dual, we develop mixed formulations of the models which are equivalent to the primal ones respectively and which are uniformly stable. We then present frameworks of designing uniformly stable mixed finite element schemes and of generating primal finite element schemes from the mixed ones. Specific finite elements are given under the frameworks as an example, and the primal scheme obtained coincides with a Durán-Liberman scheme which was constructed originally on simply-connected domains. Optimal solvers are constructed for the schemes.

The paper presents optimization and identification of the shape of elastoplastic structures. The optimization process is performed by the particle swarm method (PSO), while direct boundary value problems are solved using the parametric integral equation system (PIES). Modeling the boundary and the plastic zone in PIES is done globally by the small number of control points of parametric curves and surfaces. Such way of defining is very beneficial in comparison to so-called element methods (finite or boundary), because it reduces the number of design variables and does not enforce re-discretization during each shape change. Together with advantages of PSO it is an effective approach to solving optimization problems. There are three examples in the paper: two of identification of the shape and one in which an optimal shape is searched.

One-step forming study is a desired hull forming strategy in shipbuilding industry due to its high efficiency and economy. Compared with traditional mold forming, the multi-square punch forming (MSPF) can easily change its target shape by adjusting the height of its square punches. More importantly, MSPF provides uniformly distributed normal loading and a surface-to-surface contact between the punches and the plate, reducing plate wrinkles and dents to a great extent. This paper aims to provide solution to the one-step forming of elastic-perfectly plastic strip formed by MSPF machine. With deeper understanding of the mechanisms of the forming and the springback, accurate prediction of the springback is possible. By investigating the springback feature of the strip under MSPF, the curvature adjustment (CA) method is proposed to compensate the springback. The one-step MSPF model is then established on the basis of both the analytical model and the CA compensation method, which could properly decide the pre-designed die shape that produces the target shape of the strip. The present method is validated by published results and is proven to be in high accuracy. Case studies are also conducted by the present method to obtain the die shape of the flexible dies, which further validates the accuracy. Besides, the critical length is introduced for the particular cantilevered forming approach by the MSPF machine, and its effects on the forming accuracy and the formable range of the MSPF machine have also been discussed.