Non-overlapping Domain Decomposition Research Articles

An Optimal Robin-Type Domain Decomposition Method for Raviart-Thomas Vector Field in Three Dimensions

In this paper, we present a Robin-type nonoverlapping domain decomposition (DD) preconditioner, which is deduced from the renew equation of the Robin-Robin iteration method...

An optimal Robin-type domain decomposition method for the Well-posed Maxwell's equations in two dimensions

ABSTRACT In this paper, I present a Robin-type nonoverlapping domain decomposition (DD) preconditioner for the well-posed Maxwell's equations in two dimensions, which is deduced from the renew equation of the Robin-Robin iteration algorithm. The unknown variables to be solved in this preconditioned algebraic system are the Robin transmission data on the interface. Through choosing a suitable parameter on each subdomain boundary and using the tool of energy estimate, I prove that the condition number of the preconditioned system is optimal. Numerical results which confirm my theory are given in the last.

A multiscale direct solver for the approximation of flows in high contrast porous media

We consider a non-overlapping domain decomposition approach to approximate the solution of elliptic boundary value problems with high contrast in their coefficients. We propose a method such that initially local solutions subject to Robin boundary conditions in each primal subdomain are constructed with (locally conservative) finite element or finite volume methods. Then, a novel approach is introduced to obtain a (discontinuous) global solution in terms of linear combination of the local subdomain solutions. In the proposed algorithm the computation of local solutions for unions of subdomains are localized at nearest-neighbor subdomain boundaries, thus avoiding the solution of global interface problems. We remove discontinuities in a smoothing step that is defined on a staggered grid or dual subdomains. The resulting algorithm is naturally parallelizable and can be employed as a parallel direct solver, offering great potential for the numerical solution of large problems. In fact, subdomains can be considered small enough to fit well in GPUs and the proposed procedure can handle adaptive (in space) simulations effectively. Numerical simulations are presented and discussed. We demonstrate the effectiveness of the proposed approach with two and three dimensional high contrast and channelized coefficients, that lead to challenging approximation problems. The new procedure, although designed for parallel processing, is also of value for serial calculations.

Parallel and space-time adaptivity for the numerical simulation of cardiac action potentials

In this work, the study of parallel and space-time adaptivity for the numerical simulation of action potential propagation in cardiac electrophysiology is presented. The monodomain and bidomain models are employed for the numerical realization. This work is devoted to investigate the spatial adaptivity which is realized within multi finite element methods by using a gradient type a posteriori error estimator and the temporal adaptivity determined by a linearly implicit time integration techniques. In addition, the parallelization of such space-time adaptivity for the simulation of spiral wave dynamics based on non-overlapping domain decomposition techniques and dynamic load balancing is discussed. The numerical results demonstrate that the parallel simulations have been accelerated reasonably good using the adaptive space and time algorithm, with no significant loss in accuracy, for the single action potential wave propagation as well as reentry situations.

Exact non-overlapping domain decomposition for near-wall turbulence modeling

The near-wall non-overlapping domain decomposition (NDD) method proved to be very efficient for turbulence modeling. It represents a trade-off between accuracy and computational time. Meanwhile, the current NDD presumes the use of a simplified model near the wall. Although the approximate NDD method has successfully been applied to the Reynolds-averaged Navier–Stokes (RANS) models, it has some limitations regarding to the accuracy. In this paper, for the first time an exact NDD method is proposed, which can be applied to RANS equations without any simplification. Similar to the approximate NDD, the approach is based on a Robin-to-Dirichlet algorithm. Its convergence is proven for a multidimensional linear elliptic equation. In the new algorithm, the approximate NDD is effectively used to construct a preconditioner which immediately takes into account the near-wall flow physics. The algorithm is implemented in a low-Reynolds-number k−ε model to simulate a one-dimensional channel flow. A comparison against the approximate NDD method is provided. The results demonstrate that the approximate NDD approach performs not very well in predicting the turbulent kinetic energy and turbulent dissipation. As a contrast, the exact NDD approach can provide practically accurate prediction of the velocity field regardless of the location of the interface boundary while reducing the original computational time by one order of magnitude.

Parallel Integral Equation-Based Nonoverlapping DDM for Solving Challenging Electromagnetic Scattering Problems of Two Thousand Wavelengths

In this paper, a parallel nonoverlapping and nonconformal domain decomposition method (DDM) is proposed for fast and accurate analysis of electrically large objects in the condition of limited resources. The formulation of nonoverlapping DDM for PEC bodies is derived from combined-field integral equation (CFIE), and an explicit boundary condition is applied to ensure the continuity of electric currents across the boundary. A parallel multilevel fast multipole algorithm (MLFMA) is extended to accelerate matrix-vector multiplications of subdomains as well as the coupling between them, and the coupling between different subdomains is computed in the manner of near field to avoid the storage of the mutual impedance. An improved adaptive direction partitioning scheme is applied to the oct-tree of MLFMA to achieve high parallel efficiency. Numerical examples demonstrate that the proposed method is able to simulate realistic problems with a maximum dimension greater than 2000 wavelengths.

Fast Nonoverlapping Block Jacobi Method for the Dual Rudin--Osher--Fatemi Model

We consider nonoverlapping domain decomposition methods for the Rudin--Osher--Fatemi~(ROF) model, which is one of the standard models in mathematical image processing. The image domain is partitioned into rectangular subdomains and local problems in subdomains are solved in parallel. Local problems can adopt existing state-of-the-art solvers for the ROF model. We show that the nonoverlapping relaxed block Jacobi method for a dual formulation of the ROF model has the $O(1/n)$ convergence rate of the energy functional, where $n$ is the number of iterations. Moreover, by exploiting the forward-backward splitting structure of the method, we propose an accelerated version whose convergence rate is $O(1/n^2)$. The proposed method converges faster than existing domain decomposition methods both theoretically and practically, while the main computational cost of each iteration remains the same. We also provide the dependence of the convergence rates of the block Jacobi methods on the image size and the number of subdomains. Numerical results for comparisons with existing methods are presented.

A Finite Element Approach for the Dual Rudin--Osher--Fatemi Model and Its Nonoverlapping Domain Decomposition Methods

We consider a finite element discretization for the dual Rudin--Osher--Fatemi model using a Raviart--Thomas basis for $H_0 (\mathrm{div};\Omega)$. Since the proposed discretization has splitting property for the energy functional, which is not satisfied for existing finite difference based discretizations, it is more adequate for designing domain decomposition methods. In this paper, a primal domain decomposition method is proposed, which resembles the classical Schur complement method for the second order elliptic problems, and it achieves $O(1/n^2)$ convergence. A primal-dual domain decomposition method based on the method of Lagrange multipliers on the subdomain interfaces is also considered. Local problems of the proposed primal-dual domain decomposition method can be solved in linear convergence rate. Numerical results for the proposed methods are provided.

Explicit/implicit and Crank–Nicolson domain decomposition methods for parabolic partial differential equations

Two parallel non-overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed. The algorithms combine Crank–Nicolson scheme with implicit Galerkin finite element methods in sub-domains and explicit flux approximation along inner boundaries at each time step. Thus, parallelism can be easily achieved. L2-norm error estimates for these explicit/implicit procedures are presented, in which time step constraints are proved to be less severe than that of fully explicit schemes. Numerical experiments are also performed to verify the theoretical analysis.

Optimized Schwarz Methods for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations

Optimal control of systems governed by elliptic partial differential equations (PDEs) without constraint on the set of controls can be equivalently reformulated as a coupled system of second order elliptic PDEs, which has been considered to solve by a non-overlapping Schwarz domain decomposition method with the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions by Benamou (SIAM J Numer Anal 33:2401–2416, 1996) where a convergence analysis had been performed. Towards fast convergence of the overlapping and non-overlapping Schwarz subdomain iterations, in this paper we firstly perform, for fixed Tikhonov parameter $$\mu $$ , rigorous analyses based on optimization of the convergence factor of subdomain iterations in Fourier frequency domain to give the optimized transmission parameters involved in the above mentioned transmission conditions, as well as those involved in a Ventcell-like and a two-sided Robin-like transmission condition that we propose to accelerate the Schwarz subdomain iterations, and meanwhile we obtain also the corresponding asymptotic convergence rate estimates. The results show that the Tikhonov parameter $$\mu $$ occurs in both the optimized transmission parameters and the corresponding convergence rate estimates and affects the performance of the Schwarz domain decomposition methods significantly: when $$\mu $$ is less than a certain threshold value, with the decreasing of the Tikhonov parameter $$\mu $$ , the subdomain iteration converges more and more fast, though the regularity of the system deteriorates in this process. We lastly investigate the case where the Tikhonov parameter $$\mu =h^4$$ that is suggested by Benamou (where h is the mesh size). We obtain as well the optimized transmission parameters involved in the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions, and find that they lead to optimized Schwarz methods that are very robust in the mesh size. The analysis also sheds light on optimizing the Schwarz domain decomposition methods for biharmonic equations, since they can also be reformulated as a system of second order elliptic PDEs. We use various numerical experiments to illustrate the theoretical findings.

Interface spaces for the Multiscale Robin Coupled Method in reservoir simulation

The Multiscale Robin Coupled Method (MRCM) is a recent multiscale numerical method based on a non-overlapping domain decomposition procedure. One of its hallmarks is that the MRCM allows for the independent definition of interface spaces for pressure and flux over the skeleton of the decomposition. The accuracy of the MRCM depends on the choice of these interface spaces, as well as on an algorithmic parameter β in the Robin interface conditions imposed at the subdomain boundaries. This work presents an extensive numerical assessment of the MRCM in both of these aspects. Two types of interface spaces are implemented: usual piecewise polynomial spaces and informed spaces, the latter obtained from sets of snapshots by dimensionality reduction. Different distributions of the unknowns between pressure and flux are explored. Two non-dimensionalizations of β are tested. The assessment is conducted on realistic, high contrast, channelized permeability fields from a SPE benchmark database. The results show that β, suitably non-dimensionalized, can be fixed to unity to avoid any indeterminacy in the method. Further, with both types of spaces it is observed that a balanced distribution of the interface unknowns between pressure and flux renders the MRCM quite attractive both in accuracy and in computational cost, competitive with other multiscale methods from the literature

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.

The new mass-conserving S-DDM scheme for two-dimensional parabolic equations with variable coefficients

In the article, a new and efficient mass-conserving operator splitting domain decomposition method (S-DDM) is proposed and analyzed for solving two dimensional variable coefficient parabolic equations with reaction term. The domain is divided into multiple non-overlapping block-divided subdomains. On each block-divided subdomain, the interface fluxes are first computed explicitly by local multi-point weighted schemes and the solutions in the interior of subdomain are computed by the one-directional operator splitting implicit schemes at each time step. The scheme is proved to satisfy mass conservation over the whole domain of domain decomposition. By combining with some auxiliary lemmas and applying the energy method, we analyze theoretically the stability of our scheme and prove it to have second order accuracy in space step in the L2 norm. Numerical experiments are performed to illustrate its accuracy, conservation, stability, efficiency and parallelism. Our scheme not only keeps the excellent advantages of the non-overlapping domain decomposition and the operator splitting technique, but also preserves the global mass.

Non-overlapping domain decomposition solution schemes for structural mechanics isogeometric analysis

Isogeometric analysis (IGA) is a novel computer aided engineering technique that addresses diverse problems in computational mechanics [1–4], all under the exact geometric representation. Apart from the exact geometric representation, the high continuity of IGA shape functions enhances the accuracy and robustness of the method. However, the price to pay is that the resulting matrices are denser, with increased bandwidth and overlapping which make the solution of large-scale problems more computationally intensive. As a result, effective solution techniques are still considered an open issue for further research. In this paper, an innovative family of solution schemes based on domain decomposition methods (DDM) is proposed, that significantly reduces the computational cost. Specifically, the solution of the global system is performed with the preconditioned conjugate gradient algorithm (PCG) whose preconditioning step is evaluated with a dual DDM where special care is taken to avoid the overlapped subdomains which are inherent in decomposed IGA formulation due to the increased continuity of the shape functions.

Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE

Abstract The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard linear elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.

Multigrid methods for H(div) in three dimensions with nonoverlapping domain decomposition smoothers

SummaryWe design and analyze V‐cycle multigrid methods for an H(div) problem discretized by the lowest‐order Raviart–Thomas hexahedral element. The smoothers in the multigrid methods involve nonoverlapping domain decomposition preconditioners that are based on substructuring. We prove uniform convergence of the V‐cycle methods on bounded convex hexahedral domains (rectangular boxes). Numerical experiments that support the theory are also presented.

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in $$L^2$$ -norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.

Non-invasive global\u2013local coupling as a Schwarz domain decomposition method: acceleration and generalization

The non-invasive global–local coupling algorithm is revisited and shown to realize a simple implementation of the optimized non-overlapping Schwarz domain decomposition method. This connection is used to propose and compare several acceleration techniques, and to extend the approach to non conforming meshes.

A Domain Decomposition Scheme With Curvilinear Discretizations for Solving Large and Complex PEC Scattering Problems

A nonconformal and nonoverlapping domain decomposition (DD) scheme for surface integral equations is presented to analyze electromagnetic scattering from electrically large and complex perfect electric conducting objects. In this work, a discontinuous Galerkin approach is employed to avoid the introduction of artificial touching faces between adjacent subdomains. To weakly enforce the current continuity at the cutting contours of these subdomains, a novel interior penalty term is defined so that the error energy associated with the error charges can be minimized. Consequently, without the introduction of a stabilization term that relies on the line integral over interactions of nonmatching meshes, the proposed method provides an effective DD preconditioner and requires no effort to be extended to curvilinear elements for higher modeling accuracy. To show this, curvilinear Rao–Wilton–Glisson basis functions defined over curvilinear elements are employed in this work to demonstrate the accuracy and efficiency of the proposed DD scheme.

Parallel Integral Equation Based Nonoverlapping DDM for Solving Electrically-Large Scattering Problems in Half Space

In this paper, a parallel nonoverlapping non-conformal domain decomposition method using combined-field integral equation is proposed for fast and accurate analysis of electrically large objects situated in half space. This method decomposes the whole model into several easily solvable closed subdomains in a flexible way, and an explicit boundary condition is applied to ensure the continuity of electric currents across the boundary. The adaptive direction partitioning parallelization scheme for the oct-tree of a multilevel fast multipole algorithm is adopted to accelerate matrix-vector multiplications of subdomains as well as the coupling between them. The introducing of real image source to account for far interaction matrix greatly accelerates the process and reduces memory requirement with precision guaranteed. Numerical examples demonstrate that the proposed method is able to simulate realistic problems with a maximum dimension greater than 1000 wavelengths.