In this paper, we propose a numerical method to solve isotropic elliptic equations on point cloud by generalizing the point integral method. The idea of the point integral method is to approximate the differential operators by integral operators and discretize the corresponding integral equation on point cloud. The key step is to get the integral approximation. In this paper, with proper kernel function, we get an integral approximation for the elliptic operators with isotropic coefficients. Moreover, the integral approximation has been proved to keep the coercivity of the original elliptic operator. The convergence of the point integral method is also proved.

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes, to the correction procedure via reconstruction (CPR) framework for solving conservation laws. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper is particularly simple to implement and will not harm the conservativeness of the CPR framework. Also, it uses information only from the target cell and its immediate neighbors, thus can maintain the compactness of the CPR framework. Since the CPR framework with the WENO limiter does not in general preserve positivity of the solution, we also extend the positivity preserving limiters to the CPR framework. Numerical results in one and two dimensions are provided to illustrate the good behavior of this procedure.

In this paper, we provide a priori $L^2$ error estimates for the semi-discrete discontinuous Galerkin method and the local discontinuous Galerkin method for one- and two-dimensional nonlinear Hamilton-Jacobi equations with smooth solutions. With a special Gauss-Radau projection, the optimal error estimates on rectangular meshes are obtained.

We prove the convergence of a discontinuous Galerkin method approximating the 2-D incompressible Euler equations with discontinuous initial vorticity:$omega_0 in L^2(Omega)$. Furthermore, when $omega_0 in L^infty(Omega)$, the whole sequence is shown to be strongly convergent. This is the first convergence result in numerical approximations of this general class of discontinuous flows. Some important flows such as vortex patches belong to this class.

Wind power is an increasingly popular renewable energy. In the design process of the wind turbine blade, the accurate aerodynamic simulation is important. In the past, most of the wind turbine simulations were carried out with some low fidelity methods, such as the blade element momentum method [9]. Recently, with the rapid development of the supercomputers, high fidelity simulations based on 3D unsteady Navier-Stokes (N-S) equations become more popular. For example, Sorensen et al. studied the 3D wind turbine rotor using the Reynolds-Averaged Navier-Stokes (RANS) framework where a finite volume method and a semi-implicit method are used for the spatial and temporal discretization, respectively [17]. Bazilevs et al. investigated the aerodynamic of the NREL 5MW offshore baseline wind turbine rotor using large eddy simulation built with a deforming-spatial-domain/stabilized space-time