In this paper, we develop local discontinuous Galerkin method for the two-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in $L^{\infty}(0, T; L^{2})$ for concentration $c$, $L^{2}(0, T; L^{2})$ for $\nabla c$ and $L^{\infty}(0, T; L^{2})$ for velocity ${\bf u}$ are derived. The main techniques in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments are shown to demonstrate the theoretical results.

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Based on the recently developed data-driven time-frequency analysis [16], we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we rst run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global Intrinsic Mode Function (IMF) and instantaneous frequency. The two-level method alleviates the diculty of the mode mixing to some extent. We also present a method to reduce the end eects.

In this paper, we aim to solve one and two dimensional hyperbolic conservation laws on arbitrarily distributed point clouds. The initial condition is given on such a point cloud, and the algorithm solves for point values of the solution at later time also on this point cloud. By using the Voronoi technique and by introducing a grouping algorithm, we divide the computational domain into non-overlapping cells. Each cell is a polygon and contains a minimum number of the given points to ensure accuracy. We carefully select points in each cell during the grouping procedure, and hence are able to interpolate or fit the discrete initial values with piecewise polynomials. By adapting the traditional discontinuous Galerkin method on the constructed polygonal mesh, we obtain a stable, conservative and high order method. Numerical results for both one and two dimensional scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of our mesh generation algorithm as well as the numerical scheme.

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J-G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher-order) finite elements. This method can achieve high-order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright