Domain Decomposition Methods In Science Research Articles

Algebraic Multigrid Domain and Range Decomposition (AMG-DD/AMG-RD)

In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. To this end, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.

An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time of a highly oscillatory nature. The algorithm combines the parareal method---a parallel-in-time scheme introduced in [J.-L. Lions, Y. Maday, and G. Turinici, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), pp. 661--668]---with techniques from the heterogeneous multiscale method (cf. [W. E and B. Engquist, Notices Amer. Math. Soc., 50 (2003), pp. 1062--1070]), which make use of the slow asymptotic structure of the equations [A. J. Majda and P. Embid, Theoret. Comput. Fluid Dyn., 11 (1998), pp. 155--169]. We present error bounds, based on the analysis in [M. J. Gander and E. Hairer, in Domain Decomposition Methods in Science and Engineering XVII, Springer, Berlin, 2008, pp. 45--56] and [G. Bal, in Domain Decomposition Methods in Science and Engineering, Springer, Berlin, 2005, pp. 425--432], that demonstrate convergence of the method. A complexity analysis also demonstrates that the parallel speedup increases arbitrarily with greater scale separation. Finally, we demonstrate the accuracy and efficiency of the method on the (one-dimensional) rotating shallow water equations, which is a standard test problem for new algorithms in geophysical fluid problems. Compared to exponential integrators such as ETDRK4 and Strang splitting---which solve the stiff oscillatory part exactly---we find that we can use coarse time steps that are orders of magnitude larger (for a comparable accuracy), yielding an estimated parallel speedup of approximately 100 for physically realistic parameter values. For the (one-dimensional) shallow water equations, we also show that the estimated parallel speedup of this “asymptotic parareal method” is more than a factor of 10 greater than the speedup obtained from the standard parareal method.

Robin Schwarz Algorithm for the NICEM Method: The $\mathbf{P}_q$ Finite Element Case

In [M. J. Gander et al., in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 40, Springer-Verlag, Berlin, 2005, pp. 259--266; C. Japhet, Y. Maday, and F. Nataf, Math. Models Methods Appl. Sci., 23 (2013), pp. 2253--2292] we proposed a new nonconforming domain decomposition paradigm, the new interface cement equilibrated mortar method, based on Schwarz-type methods, that allows for the use of Robin interface conditions on nonconforming grids. The error analysis was done for $\mathbf{P}_1$ finite elements, in two and three dimensions. In this paper, we provide new numerical analysis results that allow us to extend this error analysis in two dimensions for piecewise polynomials of higher order and also prove the convergence of the iterative algorithm in all these cases.

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

A new adaptive local mesh refinement method is presented for thin film flow problems containing moving contact lines. Based on adaptation on an optimal interpolation error estimate in the Lp norm (1<p⩽∞) [L. Chen, P. Sun, J. Xu, Multilevel homotopic adaptive finite element methods for convection dominated problems, in: Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 40 (2004) 459–468], we obtain the optimal anisotropic adaptive meshes in terms of the Hessian matrix of the numerical solution. Such an anisotropic mesh is optimal for anisotropic solutions like the solution of thin film equations on moving contact lines. Thin film flow is described by an important type of nonlinear degenerate fourth order parabolic PDE. In this paper, we address the algorithms and implementation of the new adaptive finite element method for solving such fourth order thin film equations. By means of the resulting algorithm, we are able to capture and resolve the moving contact lines very precisely and efficiently without using any regularization method, even for the extreme degenerate cases, but with fewer grid points and degrees of freedom in contrast to methods on a fixed mesh. As well, we compare the method theoretically and computationally to the positivity-preserving finite difference scheme on a fixed uniform mesh which has proven useful for solving the thin film problem.