Overlapping Schwarz Methods for Unstructured Spectral Elements

A parallel multilevel domain decomposition method for source identification problems governed by elliptic equations

In this paper, we propose a parallel space-time domain decomposition method for solving an unsteadysource identification problem governed by the linear convection-diffusion equation.Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and asystem with respect to the unknown sources.The three systems have to be solved one after another.These sequential steps are not desirable for large scale parallel computing.A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization schemeto recover the time-dependent pollutant source intensity functions.We show with numerical experiments that the scheme works well with noise in the observation data.More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors,thus promising for large scale applications.

A parallel domain decomposition finite element method for massively parallel computers

We consider the numerical simulation of the left ventricle of the human heart by a hyperelastic fiber reinforced transversely isotropic model. This is an important model problem for the understanding of the mechanical properties of the human heart but its calculation is very time consuming because the lack of fast, scalable method that is also robust with respect to the model parameters. In this paper, we propose and study a fully implicit overlapping domain decomposition method on unstructured meshes for the discretized system. The algorithm is constructed within the framework of Newton–Krylov methods with an analytically constructed Jacobian. We show numerically that the algorithm is highly parallel and robust with respect to the material parameters, the large deformation, the fiber reinforcement, and the geometry of the patient-specific left ventricle. Numerical experiments show that the algorithm scales well on a supercomputer with more than 8000 processor cores.

A fully implicit domain decomposition based ALE framework for three-dimensional fluid–structure interaction with application in blood flow computation

We develop a parallel scalable domain decomposition method for the simulation of blood flows in compliant arteries in 3D, by using a fully coupled system of linear elasticity equation and incompressible Navier-Stokes equations. The system is discretized with a finite element method on unstructured moving meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping additive Schwarz method. We focus on the accuracy and parallel scalability of the algorithm, and report the parallel performance and robustness of the proposed approach by some numerical experiments carried out on a supercomputer with a large number of processors and for problems with millions of unknowns.

The analysis of transient heat conduction problems in large, complex computational domains is a problem of interest in many technological applications including electronic cooling, encapsulation using functionally graded composite materials, and cryogenics. In many of these applications, the domains may be multiply connected and contain moving boundaries making it desirable to consider meshless methods of analysis. The method of fundamental solutions along with a parallel domain decomposition method is developed for the solution of three‐dimensional parabolic differential equations. In the current approach, time is discretized using the generalized trapezoidal rule transforming the original parabolic partial differential equation into a sequence of non‐homogeneous modified Helmholtz equations. An approximate particular solution is derived using polyharmonic splines. Interfacial conditions between subdomains are satisfied using a Schwarz Neumann–Neumann iteration scheme. Outside of the first time step where zero initial flux is assumed, the initial estimates for the interfacial flux is given from the converged solution obtained during the previous time step. This significantly reduces the number of iterations required to meet the convergence criterion. The accuracy of the method of fundamental solutions approach is demonstrated through two benchmark problems. The parallel efficiency of the domain decomposition method is evaluated by considering cases with 8, 27, and 64 subdomains. Copyright 2004 © John Wiley & Sons, Ltd.

Advances on the domain decomposition solution of large scale porous media problems

In this paper, a comparison of parallel beam-tracing methods is presented and an original parallel domain decomposition method is proposed to solve numerical acoustic problems. A hybrid method between the ray-tracing and the beam-tracing method is first introduced. Then, classical parallelization methods are exposed and compared on shared and distributed memory architectures. Finally, a new parallel method based on domain decomposition principles is proposed. This method allows to handling large-scale problems better than other existing methods when taking into account the input/output and preprocessing steps. Parallel numerical experiments, carried out on a real world problem -namely the acoustic pollution analysis within a large city-illustrate the performance of this new domain decomposition method.

Coupled finite-infinite element computations are very efficient for modeling large scale acoustics problems. Parallel algorithms, like sub-structuring and domain decomposition methods, have shown to be very efficient for solving huge linear systems arising from acoustics. In this paper, a coupled finite-infinite element method is described, formulated and analyzed for parallel computations purpose. New numerical results illustrate the efficiency of this method for academic test cases and industrial problems alike.

A priori and a posteriori error analysis of hp spectral element discretization for optimal control problems with elliptic equations

Parallel domain decomposition methods for a quantum-corrected drift–diffusion model for MOSFET devices

We present parallel Schwarz-type domain decomposition preconditioned recycling Krylov subspace methods for the numerical solution of stochastic elliptic problems, whose coefficients are assumed to be a random field with finite variance. Karhunen–Loève (KL) expansion and double orthogonal polynomials are used to reformulate the stochastic elliptic problem into a large number of related but uncoupled deterministic equations. The key to an efficient algorithm lies in “recycling computed subspaces.” Based on a careful analysis of the KL expansion, we propose and test a grouping algorithm that tells us when to recycle and when to recompute some components of the expensive computation. We show theoretically and experimentally that the Schwarz preconditioned recycling GMRES method is optimal for the entire family of linear systems. A fully parallel implementation is provided, and scalability results are reported in the paper.

Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017

We study a parallel non-overlapping domain decomposition method, based on the Nesterov accelerated gradient descent, for the numerical approximation of elliptic partial differential equations. The problem is reformulated as a constrained (convex) minimization problem with the interface continuity conditions as constraints. The resulting domain decomposition method is an accelerated projected gradient descent with convergence rate [Formula: see text]. At each iteration, the proposed method needs only one matrix/vector multiplication. Numerical experiments show that significant (standard and scaled) speed-ups can be obtained.