Domain Decomposition Methods in Sciences and Engineering
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As the conference program reflects, the growing capabilities in terms of theory and available hardware allow increasingly complex non-linear and multi-physics simulations, confirming the tremendous potential and flexibility of the domain decomposition concept. Along the coarsening process, the implicit block structure of the linear system is preserved. We study this trade-off mathematically and propose an optimization strategy for the choice of the parameters, for a Backward Euler discretization of the heat equation in two dimensions. We present results for a FluidâStructure Interaction benchmark which describes self-induced oscillating deformations of an elastic beam in a flow channel. They are based on a solid theoretical foundation and shown to be scalable for many important applications. Domain decomposition methods are iterative methods for solving the often very large linear or nonlinear systems of algebraic equations that arise in various problems in mathematics, computational science, engineering and industry.

The coefficient varies over several orders of magnitude both inside of the subdomains and along the interface. Generally speaking, the most time-consuming part of the numerical simulation is the solution of the sequence of linear systems by the Krylov method. Nonlinear approaches to domain decomposition can be viewed as a strategy to localize computational work for the efficient use with future extreme-scale supercomputers. In this paper, we analyse a stabilisation technique for the so-called three-field formulation for nonoverlapping domain decomposition methods. The approach is demonstrated here for a model reaction-diffusion optimal control problem. This scheme can also be used in the computation of Maxwell eigenvalues without generating spurious eigenmodes. Our method is based on a new type of weighted average, a deluxe scaling, developed to deal with more than one variable coefficient.

Here, the partial update of the preconditioner consists in recomputing some parts of the preconditioner associated to certain subdomains, keeping the other ones frozen. With 172 participants from over 24 countries, this conference continued a long-standing tradition of internationally oriented meetings on Domain Decomposition Methods. It is therefore of interest to devise parallel solvers for such systems, and a natural idea is to apply Schwarz preconditioners to the large space-time discretized problem. Lai Factorization of the convection-diffusion operator and a possibly non overlapping Schwarz method by F. We construct B-Spline approximation spaces which are discontinuous across patch interfaces. We present a rigorous convergence analysis for the case of two non-overlapping subdomains, which shows that both methods converge in at most three iterations. This is essential for approaching peak floating point performance.

Crucial to the usage of the preconditioner are carefully chosen approximations of the 1, 1 -block and Schur complement of the saddle point system. The principal technical content of the conference has always been mathematical, but the principal motivation has been to make ef cient use of distributed memory computers for complex applications arising in science and engineering. The condition number estimates depend primarily on the Lipschitz parameters of the subdomains. In the case of warped contact boundaries and non-matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the non-penetration conditions at the contact interface. We also provide a numerical example illustrating the efficiency of the new algorithm. Alternatively, recent advancements in parallel direct solvers and the ubiquity of computer memory enables solving domain decomposition problems through the fully assembled matrix.

Tarvainen An iteration scheme for non-symmetric interface operators by C. Krause, Evaluating local approximations of the L2-orthogonal projection between non-nested finite element spaces, Tech. Russo Noniterative domain decomposition for second order hyperbolic problems by C. We also illustrate numerically that using optimized Ventcell transmission conditions leads to much faster algorithms than when using Robin transmission conditions, especially in case of anisotropic elliptic operators. In the variants proposed here, the evolution of a problem dependent global energy is controlled during the elimination process, which guarantees that the application of the nonlinear preconditioner does not increase the global energy. Time-periodic differential equations present fundamental challenges for numerical computation, because we have to consider all the discrete solutions once-in-all instead of one by one. In this paper we sketch its formulation when it is applied to second-order elliptic problems.

Thus, subdomains with warped interfaces emerge which, after an independent discretization, yield a geometrically non-conforming decomposition with small gaps and overlaps. We derive optimal transmission conditions based on an analytic factorization of the differential operator. We present the numerical analysis of the dG-IgA schemes proposed, and show a priori error estimates for geometrically matching subdomains with hanging nodes on the interface, i. While high-order finite elements often lead to solutions of higher accuracy, their associated discrete linear systems of equations are often more difficult to solve and to set up compared to those of lower order elements. An experimental analysis of the algorithms is carried out on a set of model problems. Domain decomposition methods are among the most efficient solvers for large scale applications in science and engineering.

It is well known that standard numerical schemes on uniform meshes do not yield optimal convergence rate. In both overlapping and non-overlapping cases, a full analysis of the problem is provided, and is illustrated with numerical tests. In particular, iterates of the algorithms are elements of nonlinear Sobolev spaces, and test functions are replaced by sections in certain vector bundles. Mathew A domain decomposition approach to solving the Helmholtz equation with a radiation boundary condition by O. With 172 participants from over 24 countries, this conference continued a long-standing tradition of internationally oriented meetings on Domain Decomposition Methods. The coarse space elements are well defined even for irregular subdomains, are continuous, and well suited for use in two-level or multi-level preconditioners such as overlapping Schwarz algorithms.

Preconditioners appear as bundle homomorphisms. This paper complements previous quantitative studies of transfer operators between finite element spaces associated with unrelated meshes T. Gander, Atle Loneland, Talal Rahman. Typically this happens when the fluid and solid densities are close and the fluid domain is slender, as in hemodynamical applications. Following Langer, Moore and Neumueller 2016 , we use a time-upwind test function and apply multi-patch discontinuous Galerkin IgA methodology for discretizing the evolution problem both in space and in time. It was rigorously proved last year that the preconditioner is good if the shift is at most of the size of the wavenumber. The performance of Schwarz preconditioners for elliptic problems is well understood, but how do such preconditioners perform on discretized parabolic control problems? It is expected that the new method can show a superior parallel scalability if the number of subdomains is large.

Our method is connected with a smoother based on a nonoverlapping domain decomposition method. We demonstrate the method for the simple spatially one-dimensional case. We show the performance of this approach by several numerical experiments which demonstrate the advantages of this approach. Optimal convergence rates up to an arbitrary posi- tive in both the energy norm and the L2 norm are established on graded meshes. We derive for the heat equation the convergence results for both algorithms in one spatial dimension.

A scalable three-level extension is presented, and it is validated with large scale experiments using up to 16,384 subdomains and almost a billion of degrees of freedom. A wide range of topics are discussed include hybrid formulations, Schwarz, and many more. To enlarge this time horizon, the shallow-water models of the lakes will be applied in combination with short-to-medium weather ensemble forecasts up to 15 days. We introduce a non-overlapping variant of the Schwarz algorithm for wave propagation problems with discontinuous coefficients. We present numerical results for a multigrid method employing overlapping Schwarz smoothers in various V-cycle configurations.