Mortar methods are widely used techniques for discretizations of partial differential equations and preconditioners for the algebraic systems resulting from the discretizations. For problems with high contrast and multiple scales, the standard mortar spaces are not robust, and some enrichments are necessary in order to obtain an efficient and robust mortar space. In this paper, we consider a class of flow problems in high contrast heterogeneous media, and develop a systematic approach to obtain an enriched multiscale mortar space. Our approach is based on the constructions of local multiscale basis functions. The multiscale basis functions are constructed from local problems by following the framework of the Generalized Multiscale Finite Element Method (GMsFEM). In particular, we first create a local snapshot space. Then we select the dominated modes within the snapshot space using an appropriate Proper Orthogonal Decomposition (POD) technique. These multiscale basis functions show better accuracy than polynomial basis for multiscale problems. Using the proposed multiscale mortar space, we will construct a multiscale finite element method to solve the flow problem on a coarse grid and a preconditioning technique for the fine scale discretization of the flow problem. In particular, we develop a multiscale mortar mixed finite element method using the mortar space. In addition, we will design a two-level additive preconditioner and a two-level hybrid preconditioner based on the proposed mortar space for the iterative method applied to the fine scale discretization of the flow problem. We present several numerical examples to demonstrate

A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz, and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

This paper is devoted to developing and analysing a highly accurate conservative method for solving the quantum Zakharov system. The scheme is based on a linearly-implicit compact finite difference discretization and conserve the mass as well as energy in discrete level. Detailed numerical analysis is presented which shows the method is fourth-order accurate in space and second-order accurate in time. Several numerical examples are reported to confirm the conservation properties and high accuracy of the proposed scheme. Finally the compact scheme is applied to study the convergence rate of the quantum Zakharov system to its limiting model in the semi-classical limit.

We use a suite of cosmological hydrodynamic simulations, run by two fixed grid codes, in the context of LCDM model to investigate the properties of solenoidal and dilatational, i.e, curl and compressive, motions of the intergalactic medium (IGM), and the impact of numerical viscosity on turbulence. The codes differ only in the spatial difference discretization, one is a second-order TVD scheme and the other is a fifth-order positivity-preserving WENO scheme. We find that (1) The vortical motion grows rapidly since $z=2$, and reaches $\sim 10 km/s -75 km/s$, depending on the gas density, at low redshifts. The small-scale compressive ratio $r_{CS}$ drops from $0.8$ to $0.5$ during the same period. (2) Power spectra of the solenoidal velocity possess two regimes, $\propto k^{-0.89}$ and $\propto k^{-2.02}$, while the total velocity and dilatational velocity follow the scaling $k^{-1.88}$ and $k^{-2.20}$ respectively all through the turbulent scale $\sim 2-0.2 h^{-1}Mpc$. The IGM turbulence may contain two different phases, the supersonic phase and the post-supersonic phase, simultaneously along the hierarchical structure formation history. (3) The non-thermal pressure support, measured by the vortical kinetic energy, is comparable with the thermal pressure for $\rho_b \sim 10$, or $T \sim 10^5 K$ at $z=0.25$. The deviation of the baryon fraction from the cosmic mean shows a positive correlation with the turbulence pressure support. (4) A relatively higher numerical viscosity would artificially dissipate both the compressive and vortical motions in the IGM more seriously, resulting in less developed density fluctuation and vorticity, and leading to remarkably shortened turbulence scale. Shocks in regions out of the clusters are outstandingly suppressed by the numerical viscosity since $z=2$, which could directly cause the different levels of turbulence in the IGM between the codes. We also discuss the impact of ingredients not included in our simulation.

Elliptic Bellman equations with coefficients independent of the variable x are considered. Error bounds for certain types of finite-difference schemes are obtained. These estimates are sharper than the earlier results in Krylov's article of 1997.