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The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C1 elements for velocity and C0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.

We study front speeds of curvature and strain G-equations arising in turbulent combustion. These G-equations are HamiltonJacobi type level set partial differential equations (PDEs) with non-coercive Hamiltonians and degenerate nonlinear second order diffusion. The Hamiltonian of a strain G-equation is also non-convex. Numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory (WENO) approximation of transformed G-equations on a fixed periodic domain. The advection field in the computation is a two dimensional Hamiltonian flow consisting of a periodic array of counter-rotating vortices, or cellular flows. Depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes, suitable explicit and semi-implicit time stepping methods are chosen. The turbulent flame speeds are computed as the linear growth rates of large time solutions. A

In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. In Cheng etal. (2013) a dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional structures of stochastic partial differential equations (SPDEs) and solve them efficiently. However, when the SPDEs have multiscale features in physical space, the original DyBO method becomes expensive. To address this issue, we construct multiscale basis functions within the framework of generalized multiscale finite element method (GMsFEM) for dimension reduction in the physical space. To further improve the accuracy, we also perform online procedure to construct online adaptive basis functions. In the stochastic space, we use the generalized polynomial chaos (gPC) basis functions to represent the stochastic part of the solutions. Numerical

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.