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In this paper, we give an integral approximation for the elliptic operators with anisotropic coecients on smooth manifold. Using the integral approximation, the elliptic equation is tranformed to an integral equation. The integral approximation preserves the symmetry and coercivity of the original elliptic operator. Based on these good properties, we prove the convergence between the solutions of the integral equation and the original elliptic equation.

In this paper, we propose a multi-scale discontinuous Galerkin (DG) method for second-order elliptic problems with curvilinear unidirectional rough coefficients by choosing a special non-polynomial approximation space. The key ingredient of the method lies in the incorporation of the local oscillatory features of the differential operators into the approximation space so as to capture the multi-scale solutions without having to resolve the finest scales. The unidirectional feature of the rough coefficients allows us to construct the basis functions of the DG non-polynomial approximation space explicitly, thereby greatly increasing the algorithm efficiency. Detailed error estimates for two-dimensional second-order DG methods are derived, and a general guidance on how to construct such non-polynomial basis is discussed. Numerical examples are also presented to validate and demonstrate the effectiveness of the algorithm.

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy. First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations, to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.

A fourth-order finite difference method is proposed and studied for the primitive equations (PEs) of large-scale atmospheric and oceanic flow based on mean vorticity formulation. Since the vertical average of the horizontal velocity field is divergence-free, we can introduce mean vorticity and mean stream function which are connected by a 2-D Poisson equation. As a result, the PEs can be reformulated such that the prognostic equation for the horizontal velocity is replaced by evolutionary equa-tions for the mean vorticity field and the vertical derivative of the horizontal velocity. The mean vorticity equation is approximated by a compact difference scheme due to the difficulty of the mean vorticity boundary condition, while fourth-order long-stencil approximations are utilized to deal with transport type equations for computational convenience. The numerical values for the total velocity field (both horizontal and vertical) are statically determined by a discrete realization of a differential equation at each fixed horizontal point. The method is highly efficient and is capable of produc-ing highly resolved solutions at a reasonable computational cost. The full fourth-order accuracy is checked by an example of the reformulated PEs with force terms. Addi-tionally, numerical results of a large-scale oceanic circulation are presented.