We develop numerical methods for solving partial differential equations (PDE) defined on an evolving interface represented by the grid based particle method (GBPM) recently proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems, J. Comput. Phys. 228 (2009) 77067728]. In particular, we develop implicit time discretization methods for the advectiondiffusion equation where the time step is restricted solely by the advection part of the equation. We also generalize the GBPM to solve high order geometrical flows including surface diffusion and Willmore-type flows. The resulting algorithm can be easily implemented since the method is based on meshless particles quasi-uniformly sampled on the interface. Furthermore, without any computational mesh or triangulation defined on the interface, we do not require remeshing or reparametrization in the case of highly distorted motion

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of O(k^{-1/(2^{d+1}-2)}), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of O(1/k^{0.5}).

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Numerical simulations of waves in highly heterogeneous media have important applications, but direct computations are prohibitively expensive. In this paper, we develop a new generalized multiscale finite element method with the aim of simulating waves at a much lower cost. Our method is based on a mixed Galerkin type method with carefully designed basis functions that can capture various scales in the solution. The basis functions are constructed based on some local snapshot spaces and local spectral problems defined on them. The spectral problems give a natural ordering of the basis functions in the snapshot space and allow systematically enrichment of basis functions. In addition, by using a staggered coarse mesh, our method is energy conserving and has block diagonal mass matrix, which are desirable properties for wave propagation. We will prove that our method has spectral convergence, and

For the steady-state solution of an integraldifferential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B--XF--F+ X+ XB+ X= 0, where F I-s PD, B-(b I+ s P) D-and B+ b I+ s PD+ with a nonnegative matrix P, positive diagonal matrices D, and nonnegative parameters f, b b/(1-f) and s s/(1-f). We prove the existence of the minimal nonnegative solution X under the physically reasonable assumption f+ b+ s P (D++ D-)< 1, and study its numerical computation by fixed-point iteration, Newtons method and doubling. We shall also study several special cases; eg when b = 0 and P is low-ranked, then X= s 2 UV is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.