A conforming discontinuous Galerkin finite element method was introduced by Ye and Zhang, on simplicial meshes and on polytopal meshes, which has the flexibility of using discontinuous approximation and an ultra simple formulation. The main goal of this paper is to improve the above discontinuous Galerkin finite element method so that it can handle nonhomogeneous Dirichlet boundary conditions effectively. In addition, the method has been generalized in terms of approximation of the weak gradient. Error estimates of optimal order are established for the corresponding discontinuous finite element approximation in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented to confirm the theory.

With the rapid development of computer technology, numerical simulation has become the third scientific research tool besides theoretical analysis and experimental research. As the core of numerical simulation, constructing efficient, accurate and stable numerical methods to simulate complex scientific and engineering problems has become a key issue in computational mechanics. The article outlines the application of singular boundary method to the large-scale and high-frequency acoustic problems. In practical application, the key issue is to construct efficient and accurate numerical methodology to calculate the large-scale and high-frequency sound field. This article focuses on the following two research areas. They are how to discretize partial differential equations into more appropriate linear equations, and how to solve linear equations more efficiently. The bottleneck problems encountered in the computational acoustics are used as the technical routes, i.e., efficient solution of dense linear system composed of ill-conditioned matrix and stable simulation of wave propagation at low sampling frequencies. The article reviews recent advances in emerging applications of the singular boundary method for computational acoustics. This collection can provide a reference for simulating other more complex wave propagation.

Semi-discrete and fully discrete mixed finite element methods are considered for Maxwell-model-based problems of wave propagation in linear viscoelastic solid. This mixed finite element framework allows the use of a large class of existing mixed conforming finite elements for elasticity in the spatial discretization. In the fully discrete scheme, a Crank-Nicolson scheme is adopted for the approximation of the temporal derivatives of stress and velocity variables. Error estimates of the semi-discrete and fully discrete schemes, as well as an unconditional stability result for the fully discrete scheme, are derived. Numerical experiments are provided to verify the theoretical results.

In this paper, we propose a conforming finite element method coupling penalty method for the linearly elastic flexural shell to overcome computational difficulties. We start with discretizing the displacement variable, i.e., the two tangent components of the displacement are discretized by using conforming finite elements (linear element), and the normal component of the displacement is discretized by using conforming Hsieh-Clough-Tocher element (HCT element). Then, the existence, uniqueness, stability, convergence and a priori error estimate of the corresponding analyses are proven and analyzed. Finally, we present numerical experiments with a portion of the conical shell and a portion of the cylindrical shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme.

In this paper, a numerical method for the electroneutral micro-fluids based on the finite element method will be given. In order to deal with the non-linearity of the equation, the modified characteristics method will be used to deal with the temporal derivative term and the convective term. In this way, the non-linear equation can be linearized. Then, we will give the unconditional stability and optimal error estimation. At last, some numerical results are given to show the effectiveness of our method. From the stability analysis we can see that the method is unconditionally stable. The numerical results show that our method is robust.

Based on newly developed weight-based smoothness detectors and nonlinear interpolations designed to capture discontinuities for the multiderivative combined dissipative compact scheme (MDCS), hybrid linear and nonlinear interpolations are proposed to form hybrid MDCS. These detectors are derived from the weights used for the nonlinear interpolations and can provide suitable switches between the linear and the nonlinear schemes to realize the characteristics for the hybrid MDCS of capturing discontinuities and maintaining high resolution in the region without large discontinuities. To save computational cost, the nonlinear scheme with characteristic decomposition is only applied in the detected discontinuities region by specially designed hybrid strategy. Typical tests show that the hybrid MDCS is capable of capturing discontinuities and maintaining high resolution power for the smooth region at the same time. With the satisfaction of the geometric conservative law (GCL), the MDCS is further applied on curvilinear mesh to present its promising capability of handling pragmatic simulations.

In recent years, various kinds of cloak devices were designed by transformation optics, but these cloak metamaterials are anisotropic and difficult to manufacture. In this paper, we designed the isotropic cloak metamaterials based on the numerical method of the optimization theory to the inverse medium problem. This method has universality, and it is not limited by the shape and type of the cloak device. This isotropic material is easier to manufacture in practice than anisotropic material. A large number of numerical results show the effectiveness of the method.

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state constraints. The control problem is approximated by the $hp$ spectral element method with high accuracy and geometric flexibility. Optimality conditions of the continuous and discrete optimal control problems are presented, respectively. The a posteriori error estimates both for the control and state variables are established in detail. In addition, illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

The Peng-Robinson equation of state, one of the most extensively applied equations of state in the petroleum industry and chemical engineering, has an excellent appearance in predicting the thermodynamic properties of a wide variety of materials. It has been a great challenge on how to design numerical schemes with preservation of mass conservation and energy dissipation law. Based on the exponential time difference combined with the stabilizing technique and added Lagrange multiplier enforcing the mass conservation, we develop the efficient first- and second-order numerical schemes with preservation of maximum bound principle (MBP) to solve the single-component two-phase diffuse interface model with Peng-Robinson equation of state. Convergence analyses as well as energy stability are also proven. Several two-dimensional and three-dimensional experiments are performed to verify these theoretical results.

In this paper we present how nonlinear stochastic Itô differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct manifold. To this end, we formulate an approach based on Runge-Kutta–Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds. 

Moreover, we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.