We propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry. By introducing a generalized vorticity-stream formulation, the divergence free constraints are automatically satisfied. In addition, with explicit treatment of the nonlinear terms and local vorticity boundary condition, the Navier–Stokes (MHD, respectively) equation essentially decouples into 2 (4, respectively) scalar equation and thus the scheme is very efficient. Moreover, with proper discretization of the nonlinear terms, the scheme preserves both energy and helicity identities numerically. This is achieved by recasting the nonlinear terms (convection, vorticity stretching, geometric source, Lorentz force and electro-motive force) in terms of Jacobians. This conservative property is valid even in the presence of the pole singularity for axisymmetric flows. The exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity. Numerical examples have demonstrated both accuracy and efficiency of the scheme. Finally, local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost.

We consider polynomial approximation on the unit sphere $\mathbb{S}^2=\{(x,y,z)\in \mathbb{R}^3:x^2+y^2+z^2=1\}$ by a class of regularized discrete least squares methods with novel choices for the regularization operator and the point sets of the discretization. We allow different kinds of rotationally invariant regularization operators, including the zero operator (in which case the approximation includes interpolation, quasi-interpolation, and hyperinterpolation); powers of the negative Laplace--Beltrami operator (which can be suitable when there are data errors); and regularization operators that yield filtered polynomial approximations. As node sets we use spherical $t$-designs, which are point sets on the sphere which when used as equal-weight quadrature rules integrate all spherical polynomials up to degree $t$ exactly. More precisely, we use well conditioned spherical $t$-designs obtained in a previous paper by maximizing the determinants of the Gram matrices subject to the spherical design constraint. For $t\ge 2L$ and an approximating polynomial of degree $L$ it turns out that there is no linear algebra problem to be solved and the approximation in some cases recovers known polynomial approximation schemes, including interpolation, hyperinterpolation, and filtered hyperinterpolation. For $t \in [L, 2L)$ the linear system needs to be solved numerically. Finally, we give numerical examples to illustrate the theoretical results and show that well chosen regularization operator and well conditioned spherical $t$-designs can provide good polynomial approximation on the sphere, with or without the presence of data errors.

Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction. Graph Laplacian is one widely used discrete laplacian on point cloud. It was previously proved by Belkin and Niyogi [4] that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of innitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Point Integral method (PIM) [15] to solve elliptic equations and corresponding eigenvalue problem on point clouds. In this paper, we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of innitely many random samples. Moreover, one estimate of the rate of the convergence is given.

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In this paper, we investigate the superconvergence property of the local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time dependent fourth order problems. We prove that the error between the LDG solution and a particular projection of the exact solution, $\bar{e}_u$, achieves $(k+\frac32)$-th order superconvergence when polynomials of degree $k$ ($k\ge 1$) are used. Numerical experiments of $P^k$ polynomials, with $1\le k \le3$, are displayed to demonstrate the theoretical results, which show that the error $\bar{e}_u$ actually achieves $(k+2)$-th order superconvergence, indicating that the error bound for $\bar{e}_u$ obtained in this paper is sub-optimal. Initial-boundary value problems, nonlinear equations and solutions having singularities are numerically investigated to verify that the conclusions hold true for very general cases.