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The Schwarz alternating method can be used to solve linear elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approx- imated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers four Schwarz alternating methods for the N-dimensional, steady, viscous, incompressible Navier­Stokes equations, N ≤ 4. It is shown that the Schwarz sequences converge to the true solution provided that the Reynolds number is sufficiently small. [ABSTRACT FROM AUTHOR]

A successive continuation method for locating connecting orbits in parametrized systems of autonomous ODEs was considered in [Numer. Algorithms, 14 (1997), pp. 103­124]. In this paper we present an improved algorithm for locating and continuing connecting orbits, which includes a new algorithm for the continuation of invariant subspaces. The latter algorithm is of independent interest and can be used in contexts different than the present one. [ABSTRACT FROM AUTHOR]

We present a method for creating large two-dimensional graded Delaunay meshes on parallel machines. The method decouples the mesh generation procedure for each subdomain and thus eliminates the communication between processors, while the final mesh is of the same guaranteed quality as in the sequential case. This work extends our previous result on the uniform Delaunay decoupling method [L. Linardakis and N. Chrisochoides, SIAM J. Sci. Comput., 27 (2006), pp. 1394- 1423], by allowing the element size to be governed by a sizing function, or a background mesh. The graded Delaunay decoupling method can produce in parallel very large graded meshes efficiently, demonstrating high speedup and scalability. Ten billion elements can be created in five minutes with small overrefinement (about 2%). The decoupling parallelization approach is effective, using off-the-shelf, well-tested, and fine-tuned sequential mesh generation libraries without modification. [ABSTRACT FROM AUTHOR]

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is used to show that in a nonlinear case the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem which involves the tangent linear model constraints. The inverse Hessian is constructed by the quasi-Newton BFGS algorithm when solving the auxiliary data assimilation problem. A fully nonlinear ensemble procedure is developed to verify the accuracy of the proposed algorithm. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

A propagating interface can develop corners and discontinuities as it advances. Level set algorithms have been extensively applied for the problems in which the solution has advancing fronts. One of the most popular level set algorithms is the so-called fast marching method (FMM), which requires total O(N log2 N) operations, where N is the number of grid points. The article is concerned with the development of an O(N) level set algorithm called the group marching method (GMM). The new method is based on the narrow band approach as in the FMM. However, it is incorporating a correction-by-iteration strategy to advance a group of grid points at a time, rather than sorting the solution in the narrow band to march forward a single grid point. After selecting a group of grid points appropriately, the GMM advances the group in two iterations for the cost of slightly larger than one iteration. Numerical results are presented to show the efficiency of the method, applied to the eikonal equation in two and three dimensions. [ABSTRACT FROM AUTHOR]