Non-overlapping Domain Decomposition Method Research Articles

Convergence analysis and applicability of a domain decomposition method with nonlocal interface boundary conditions

In the past, the domain decomposition method was developed successfully for solving large-scale linear systems. However, the problems with significant nonlocal effect remain a major challenger for applying the method efficiently. In order to sort out the problem, a non-overlapping domain decomposition method with nonlocal interface boundary conditions was recently proposed and studied both theoretically and numerically. This paper is the report on the further development of the method, aiming to provide a comprehensive convergence analysis of the method, with supplementary numerical tests to support the theoretical result. The nonlocal effect of the problem is found to be reflected in both the governing equation and boundary conditions, and the effect of the latter was never taken into account, although playing a significant role in affecting the convergence. In addition, the paper extends the applicability of the analysis result drawn from the Poisson’s equation to more complicated problems by examining the symbols of the Steklov–Poincaré operators. The extended application includes a model equation arising from fluid dynamics and the high performance of the domain decomposition method in solving this equation is better elaborated.

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

A family of Nonoverlapping Spectral Additive Schwarz methods (NOSAS) and their economic versions

NOSAS (Nonoverlapping Spectral Additive Schwarz) methods are nonoverlapping iterative domain decomposition methods for efficiently solving large sparse linear systems arising from elliptic problems with heterogeneous coefficients. NOSAS methods are of additive Schwarz types with local Dirichlet solver on each nonoverlapping subdomain Ωi, plus a coarse problem on the interface of the subdomains. The coarse problem has global and local interaction components which are associated respectively to low-frequency and high-frequency modes obtained locally from a generalized eigenvalue problem defined on each subdomain’s interface. In this paper, we propose several variants of the NOSAS methods by selecting different generalized eigenvalue problems and different coarse space extensions. Additionally, we propose an economic variant to reduce the complexity of these generalized eigenvalue problems. The analysis of all the variants is established, and numerical tests are provided.

Investigation of combustion model via the local collocation technique based on moving Taylor polynomial (MTP) approximation/domain decomposition method with error analysis

In this paper, we develop a new meshless numerical procedure for simulating the combustion model. To that end, we employ a local meshless collocation method according to the moving Taylor polynomial (MTP) approximation. The space derivative is approximated by using the local approach and then the Crank–Nicolson algorithm is utilized to approximate the time derivative. The stability and convergence of the time-discrete formulation are discussed, analytically and numerically. The Broyden method is applied to solve this nonlinear system. Since the size of the physical domain is large, we employ the non-overlapping domain decomposition method (DDM) to obtain a faster numerical algorithm. The local meshless approaches are efficient numerical techniques to simulate models in the fluid flow. The obtained results show that the proposed numerical formulation has efficient results for solving this mathematical model.

Adaptive fail-safe topology optimization using a hierarchical parallelization scheme

This work presents an efficient, flexible, and scalable strategy to implement density-based topology optimization formulation for fail-safe structural design. Such optimized designs can operate after faults modeled as loss of stiffness. However, the need for assessing the physical behavior in all the fault cases can make this problem unfeasible computationally. We combine many ingredients to exploit the different levels of parallelism of the formulation to deal successfully with this problem. We use a non-overlapping domain decomposition method to solve the failure cases using multi-core computing with distributed memory computation. These subdomains use adaptive mesh refinement (AMR) techniques to focus computing efforts on the regions of interest. A key point to evaluate the fault cases is the organization of the computing threads, especially in clusters of computers. We group the computing threads by physical computing nodes to reduce the inter-node communications, maximizing intra-node communications to mitigate bandwidth problems. Another ingredient of paramount importance is the use of computing buffers to adapt the evaluation of the fault cases to the computing resources. We test the scalability of the computational framework using a large cluster of computers.

Fast Non-overlapping Domain Decomposition Methods for Continuous Multi-phase Labeling Problem

Fast Non-overlapping Domain Decomposition Methods for Continuous Multi-phase Labeling Problem

A locally optimal preconditioned Newton-Schur method for symmetric elliptic eigenvalue problems

A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincaré operator is used to project the eigenvalue problem on the domain Ω \Omega onto the nonlinear eigenvalue subproblem on Γ \Gamma , which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on Γ \Gamma . Four different strategies are proposed to build the hierarchical subspace U k + 1 U_{k+1} over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace U k + 1 U_{k+1} . The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be γ = 1 − c 0 T h , H − 1 \gamma =1-c_{0}T_{h,H}^{-1} , where c 0 c_{0} is a constant independent of the fine mesh size h h , the coarse mesh size H H and jumps of the coefficients; whereas T h , H T_{h,H} is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.

A Space-Time Multiscale Mortar Mixed Finite Element Method for Parabolic Equations

We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with nonmatching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability as well as a priori error estimates for the spatial and temporal errors are established. A space-time nonoverlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time.

Fast finite element electrostatic analysis with domain decomposition method

Accurate simulation of electrostatic field in electronic devices is very important to improve their performance. When the traditional finite element method is used to analyze the electrostatic field of large and complex structures, the calculation speed is slow and the computing resources are greatly consumed. Therefore, in this paper, we investigate electrostatic analysis with a non-overlapping domain decomposition method based on the scalar finite element method under a novel transmission condition. In order to improve the efficiency of matrix solving and reduce the memory consumption, the block Jacobi preconditioner is introduced. In addition, the multifrontal block incomplete Cholesky preconditioner is utilized to further improve the computational performance based on the block Jacobi preconditioner. Several numerical examples are simulated and compared with the analytical solutions and the commercial software Maxwell 3D. The results demonstrate the accuracy and efficiency of the proposed domain decomposition method.

Convergence Analysis of Newton–Schur Method for Symmetric Elliptic Eigenvalue Problem

In this paper, we consider the Newton–Schur method in Hilbert space and obtain quadratic convergence. For the symmetric elliptic eigenvalue problem discretized by the standard finite element method and nonoverlapping domain decomposition method, we use the Steklov–Poincaré operator to reduce the eigenvalue problem on the domain into the nonlinear eigenvalue subproblem on , which is the union of subdomain boundaries. We prove that the convergence rate for the Newton–Schur method is , where the constant is independent of the fine mesh size and coarse mesh size , and and are errors after and before one iteration step, respectively. For one specific inner product on , a sharper convergence rate is obtained, and we can prove that . Numerical experiments confirm our theoretical analysis.

Multiphysics Computing of Challenging Antenna Arrays Under a Supercomputer Framework

A parallel multiphysics simulation solver is developed to solve electromagnetic-thermal-mechanical coupling for some challenging large-scale antenna arrays. To achieve high scalability of supercomputer architectures, we reconstruct the preconditioned BiCGSTAB method and the non-overlapping domain decomposition method, so that the most resource-intensive matrix factorization steps can be performed in parallel independently within subdomains. The electromagnetic and thermal fields are solved separately, while coupled through the dissipated power and the temperature-dependent material parameters; after thermal steady state is reached, the mechanical simulation is stimulated subject to the temperature rise. The accuracy of electromagnetic-thermal coupling and thermal stress solution are first validated, and then the strong/weak parallel scalability experiments of the developed multiphysics solver are performed on supercomputer. Finally, an extremely challenging antenna array is simulated using the proposed solver, where to our best knowledge we bring the scale of multiphysics simulations excited by frequency-domain electromagnetic fields to the order of billion unknowns for the first time.

A computational slip boundary condition for near-wall turbulence modeling

Near-wall turbulence modeling represents one of challengers in the computational fluid dynamics. To tackle this problem, the near-wall non-overlapping domain decomposition (NDD) method proved to be very efficient. It has been successfully used with different Reynolds-averaged Navier–Stokes models. In NDD the computational domain is split into two non-overlapping sub-domains: an inner region near the wall, which is characterized by high gradients, and the outer region. To simplify the solution, in the inner region the thin-layer model can be used. In this case, NDD represents a trade-off between the accuracy and computational time. It has been demonstrated on numerous test cases that NDD is able to save up to one order of computational time while retaining practically high accuracy. A practical drawback of the algorithm is a need to split the computational domain into two regions. In the present paper, for the first time the NDD is realized implicitly. For this purpose, specific boundary conditions of Robin type are derived at the wall. Such boundary conditions reduce the gradients of the solution near the wall. The key property is that the solution can be obtained on a relatively coarse grid and then be recalculated in the inner region. In the approach it is guaranteed that the original outer and updated inner solutions are linked smoothly. The algorithm with the new boundary conditions can be easily implemented in standard codes. This is demonstrated with the code OpenFOAM. In addition, in the paper a more accurate approach to obtain the solution in the inner region is realized. The efficiency of the entire technique is demonstrated on test cases with modeling turbulent flows in a channel and asymmetric diffuser.

A Well-Conditioned Weak Coupling of Boundary Element and High-Order Finite Element Methods for Time-Harmonic Electromagnetic Scattering by Inhomogeneous Objects

The aim of this paper is to propose efficient weak coupling formulations between the boundary element method and the high-order finite element method for solving time-harmonic electromagnetic scattering problems. The approach is based on the use of a non-overlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Pade approximants of the nonlocal Magnetic-to-Electric operators. Numerical results are presented to validate and analyze the new approach for both homogeneous and inhomogeneous scatterers.

A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation

It is well-known that the convergence rate of non-overlapping domain decomposition methods (DDMs) applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly depends on the transmission condition enforced at the interfaces between the subdomains. Transmission operators based on perfectly matched layers (PMLs) have proved to be well-suited for configurations with layered domain partitions. They are shown to be a good compromise between basic impedance conditions, which can lead to slow convergence, and computational expensive conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care.In this work, we present a non-overlapping substructured DDM with PML transmission conditions for checkerboard (Cartesian) decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. Two possible Lagrange multiplier finite element spaces are presented, and the behavior of the corresponding DDM is analyzed on several numerical examples.

Spectral Coarse Spaces for the Substructured Parallel Schwarz Method

The parallel Schwarz method (PSM) is an overlapping domain decomposition (DD) method to solve partial differential equations (PDEs). Similarly to classical nonoverlapping DD methods, the PSM admits a substructured formulation, that is, it can be formulated as an iteration acting on variables defined exclusively on the interfaces of the overlapping decomposition. In this manuscript, spectral coarse spaces are considered to improve the convergence and robustness of the substructured PSM. In this framework, the coarse space functions are defined exclusively on the interfaces. This is in contrast to classical two-level volume methods, where the coarse functions are defined in volume, though with local support. The approach presented in this work has several advantages. First, it allows one to use some of the well-known efficient coarse spaces proposed in the literature, and facilitates the numerical construction of efficient coarse spaces. Second, the computational work is comparable or lower than standard volume two-level methods. Third, it opens new interesting perspectives as the analysis of the new two-level substructured method requires the development of a new convergence analysis of general two-level iterative methods. The new analysis casts light on the optimality of coarse spaces: given a fixed dimension m, the spectral coarse space made by the first m dominant eigenvectors is not necessarily the minimizer of the asymptotic convergence factor. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

Domain decomposition and partitioning methods for mixed finite element discretizations of the Biot system of poroelasticity

We develop non-overlapping domain decomposition methods for the Biot system of poroelasticity in a mixed form. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. In the monolithic method, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. We show that the resulting interface operator is positive definite and analyze the convergence of the iteration. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. Numerical experiments are presented to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency.

BDDC algorithms for advection-diffusion problems with HDG discretizations

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory.

Convergence Analysis of the Continuous and Discrete Non-overlapping Double Sweep Domain Decomposition Method Based on PMLs for the Helmholtz Equation

Convergence Analysis of the Continuous and Discrete Non-overlapping Double Sweep Domain Decomposition Method Based on PMLs for the Helmholtz Equation

SIE-DDM With Higher Order Hierarchical Vector Basis Functions for Solving Electromagnetic Problems on Multiscale Metallic Targets

In this article, a nonconformal and nonoverlapping domain decomposition method (DDM) based on surface integral equation (SIE) is proposed for the electromagnetic (EM) scattering or radiation of multiscale metallic targets. To reduce the unknown amount of the conventional SIE for multiscale EM simulation, we apply curved triangular elements and higher order hierarchical vector (HOHV) basis functions to the SIE. Next, to increase the flexibility of geometrical modeling and accelerate the convergence of the presented SIE system for electrically large and multiscale metallic targets, a DDM scheme is further developed to employ a discontinuous Galerkin (DG) approach to glue conformal/nonconformal surface grids between adjacent subdomains. In addition, an interior penalty term is introduced to stabilize the DDM solution, and half edge-based HOHV basis functions are introduced to model the current associated with nonconformal surface elements. Meanwhile, the basis expansion and recombination (BER) technique is introduced to significantly accelerate the matrix-filling and improve the efficiency. The flexibility of basis order selection is further enhanced by the hierarchical characteristic of HOHV bases. Finally, several numerical results are provided to demonstrate the accuracy, efficiency, and flexibility of the proposed HO-DG-DDM.

Solving the forward-backward heat equation with a non-overlapping domain decomposition method based on multiquadric RBF meshfree method

‎‎‎‎‎In this paper, we present a numerical technique to deal with the one-dimensional forward-backward heat equations. First, the physical domain is divided into two non-overlapping subdomains resulting in two separate forward and backward subproblems, and then a meshless method based on multiquadric radial basis functions is employed to treat the spatial variables in each subproblem using the Kansa’s method. We use a time discretization scheme to approximate the time derivative by the forward and backward finite difference formulas. In order to have adequate boundary conditions for each subproblem, an initial approximate solution is assumed on the interface boundary, and the solution is improved by solving the subproblems in an iterative way. The numerical results show that the proposed method is very useful and computationally efficient in comparison with the previous works.