We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method [L. Pareschi and G. Russo, SIAM J. Numer. Anal., 37 (2000), pp. 1217–1245; I. M. Gamba and S. H. Tharkabhushanam, J. Comput. Phys., 228 (2009), pp. 2012–2036], which requires O(N6) memory to store precomputed weights and has O(N6) numerical complexity, the new method has complexity O(MN4 log N), where N is the number of discretization points in each of the three velocity dimensions and M is the total number of discretization points on the sphere and M ≪ N2. Furthermore, it requires no precomputation for the variable hard sphere model and only O(MN4) memory to store precomputed functions for more general collision kernels. Although a faster spectral method is available [C. Mouhot and L. Pareschi, Math. Comp., 75 (2006), pp. 1833–1852] (with complexity O(MN3 logN)), it works only for hard sphere molecules, thus limiting its use for practical problems. Our new method, on the other hand, can apply to arbitrary collision kernels. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method.

Abstract BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance T O L. Upper bounds for condition numbers of the preconditioned systems are shown to be C T O L, with the constant C depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.

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

A novel discontinuous Galerkin (DG) method is developed to solve timedependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are L^2 stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal L^2 error estimate of O(h^{k+1}) for polynomials of degree k for semi-discrete DG schemes, and the L2 error of O(h^{k+1} + (\Delta t)^2) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift–Hohenberg equation endowed with a decay free energy is presented.

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the Łojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (Hölder type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings.