In this paper, we consider an online basis enrichment mixed generalized multiscale method with oversampling, for solving flow problems in highly heterogeneous porous media. This is an exten-sion of the online mixed generalized multiscale method [6]. The multiscale online basis functions are computed by solving a Neumann problem in an over-sampled domain, instead of a standard neighborhood of a coarse face. We are motivated by the restricted domain decomposition method. Extensive numerical experiments are presented to demonstrate the performance of our methods for both steady-state flow, and two-phase flow and transport problems.

In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree $k$ and suitable initial discretizations, the DG solution is $(2k+1)$-th order accurate at the downwind points and $(k+2)$-th order accurate at all the other downwind-biased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of $k+1$ at all the interior upwind-biased Radau points. Besides the above, the DG solution is also $(k+2)$-th order accurate towards a particular projection of the exact solution and the numerical cell averages are $(2k+1)$-th order accurate. Numerical experiments are presented to confirm the theoretical results.

We consider the reconstruction of the Robin impedance coefficient of a heat conduction system in a two-dimensional spatial domain from the time-average measurement specified on the boundary. By applying the potential representation of a solution, this nonlinear inverse problem is transformed into an ill-posed integral system coupling the density function for potential and the unknown boundary impedance. The uniqueness as well as the conditional stability of this inverse problem is established from the integral system. Then we propose to find the boundary impedance by solving a non-convex regularizing optimization problem. The well-posedness of this optimization problem together with the convergence property of the minimizer is analyzed. Finally, based on the singularity decomposition of the potential representation of the solution, two iteration schemes with their numerical realizations are proposed to solve this optimization problem

We investigate the angular distribution of Ly$\alpha$ photons transferring in or emergent from an optically thick medium. Since the evolutions of specific intensity $I$ in the frequency space and the angular space are coupled with each other, we first develop the WENO numerical solver in order to find the time-dependent solutions of the integro-differential equation of $I$ in frequency and angular space simultaneously. We first show that the solutions with the Eddington approximation, which assume $I$ to be linearly dependent on the angular variable $\mu$, yield similar frequency profiles of the photon flux as that without the Eddington approximation. However, the solutions of the $\mu$ distribution evolution are significantly different from that given by Eddington approximation. First, the angular distribution of $I$ are found to be substantially dependent on the frequency of photons. For photons with the resonant frequency $\nu_0$, $I$ contains only a linear term of $\mu$. For photons with frequency at the double peaks of the flux, the $\mu$-distribution is highly anisotropic, in which most photons are in the direction of radial forward. Moreover, either at $\nu_0$ or at the double peaks, the $\mu$ distributions actually are independent of the initial $\mu$ distribution of photons of the source. This is because the photons with frequency either of $\nu_0$ or of the double peaks undergo the process of forgetting their initial conditions due to the resonant scattering. We also show that the optically thick medium is a collimator of photons at the double peaks. Photons from the double peaks form a forward beam with very small spread angle.

nthispaperwewillpresentthelocalstabilityanalysisandlocalerrorestimatefor the local discontinuous Galerkin (LDG) method, when solving the time-dependent singularly perturbed problems in one dimensional space with a stationary outflow boundary layer. Based on a general framework on the local stability, we obtain the optimal error estimate out of the local subdomain, which is nearby the outflow boundary point and has the width of O(h log(1/h)), for the semi-discrete LDG scheme and the fully-discrete LDG scheme with the second order explicit Runge–Kutta time-marching. Here h is the maximum mesh length. The numerical experiments are given also.