We propose a new and simple discretization, named the modified virtual grid difference (MVGD), for numerical approximation of the Laplace--Beltrami (LB) operator on manifolds sampled by point clouds. The key observation is that both the manifold and a function defined on it can both be parametrized in a local Cartesian coordinate system and approximated using least squares. Based on the above observation, we first introduce a local virtual grid with a scale adapted to the sampling density centered at each point. Then we propose a modified finite difference scheme on the virtual grid to discretize the LB operator. Instead of using the local least squares values on all virtual grid points like the typical finite difference method, we use the function value explicitly at the grid located at the center (coinciding with the data point). The new discretization provides more diagonal dominance to the resulting linear system and

It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws (\RefOne{Jiang and Shu (1994)} \cite{jiang1994cell}) and symmetric hyperbolic systems (\RefOne{Hou and Liu (2007)} \cite{hou2007solutions}), in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and / or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in \RefOne{Carpenter et al (2014)} \cite{carpenter2014entropy} and \RefOne{Gassner (2013)} \cite{gassner2013skew}. The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the \RefThree{entropy conservative flux} within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in [11] which is based on the FischerBurmeister function. We investigate this continuation approach again by using a more general function, called the generalized FischerBurmeister function. However, the theoretical background for such extension can not be easily carried over. Indeed, it needs some subtle analysis.

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.

We show that the gradient mapping of the squared norm of FischerBurmeister function is globally Lipschitz continuous and semismooth, which provides a theoretical basis for solving nonlinear second-order cone complementarity problems via the conjugate gradient method and the semismooth Newtons method.