Non-overlapping Domain Decomposition Method Research Articles

A Robin-type Domain Decomposition Method with Red-Black Partition

In this paper, we present a new Robin-type nonoverlapping domain decomposition (DD) preconditioner. The unknown variables to be solved in this preconditioned algebraic system are the Robin transmission data on the interface, which are different from the well-known DD methods like substructuring nonoverlapping DD method and FETI method. Through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate, for the second-order elliptic problem, we prove that the condition number of the preconditioned system is $C(1+\log(\frac{H}{h}))^2$, where $H$ is the coarse mesh size and $h$ is the fine mesh size. Moreover, in this paper, we always assume that there is a red-black partition for the whole domain $\Omega$. Numerical results are given to illustrate the efficiency of our DD preconditioner.

A D-N Alternating Algorithm for Solving 3D Exterior Helmholtz Problems

The nonoverlapping domain decomposition method, which is based on the natural boundary reduction, is applied to solve the exterior Helmholtz problem over a three-dimensional domain. The basic idea is to introduce a spherical artificial boundary; the original unbounded domain is changed into a bounded subdomain and a typical unbounded region; then, a Dirichlet-Nuemann (D-N) alternating method is presented; the finite element method and natural boundary element methods are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. The convergence of the D-N alternating algorithm and its discretization are studied. Some numerical experiments are presented to show the performance of this method.

A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods

This paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non-overlapping domain decomposition method (FETI or BDD). We propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization. Numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solver.

Fast 3-D Thermal Simulation for Integrated Circuits With Domain Decomposition Method

For accurate thermal simulation of integrated circuits (ICs), heat sink components in chip package must be considered. In this letter, techniques based on the domain decomposition method (DDM) are presented for the 3-D thermal simulation of nonrectangular IC thermal model including heat sink and heat spreader. A relaxed nonoverlapping DDM algorithm is employed to convert the problem to subproblems on rectangular subdomains. Then, a nonconformal discretization strategy is proposed to reduce the problem complexity with negligible error. Numerical experiments on several 2-D and 3-D IC test cases demonstrate that the relaxed nonoverlapping DDM is faster than the other preconditioned conjugate gradient algorithms with same mesh grid. The nonconformal discretization achieves further 10× reduction of runtime and memory usage.

Performance study of domain decomposition methods for 2D parallel finite element analysis

The aim of this paper is to give a unified comparison of non-overlapping domain decomposition methods for solving magnetic field problems. The methods under investigation are the Schur complement method and the Lagrange multiplier based finite element tearing and interconnecting method, and their solvers. The performance of these methods has been investigated in detail for two-dimensional magnetic field problems as case studies.

Domain decomposition for coupled Stokes and Darcy flows

A non-overlapping domain decomposition method is presented to solve a coupled Stokes–Darcy flow problem in parallel by partitioning the computational domain into multiple subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling is based on appropriate interface matching conditions. The global problem is reduced to an interface problem by eliminating the interior subdomain variables. The interface problem is solved by an iterative procedure, which requires solving subdomain problems at each iteration. Finite element techniques appropriate for the type of each subdomain problem are used to discretize it. The condition number of the resulting algebraic system is analyzed and numerical tests verifying the theoretical estimates are provided.

비중첩 영역 분할기법 기반 병렬해석의 정확도 분석

In this paper, an accuracy analysis of parallel method based on non-overlapping domain decomposition method is carried out. In this approach, proposed by Tak et al.(2013), the decomposed subdomains do not overlap each other and the connection between adjacent subdomains is determined via simple connective finite element named interfacial element. This approach has two main advantages. The first is that a direct method such as gauss elimination is available even in a singular problem because the singular stiffness matrix from floating domain can be converted to invertible matrix by assembling the interfacial element. The second is that computational time and storage can be reduced in comparison with the traditional finite element tearing and interconnect(FETI) method. The accuracy of analysis using proposed method, on the other hand, is inclined to decrease at cross points on which more than three subdomains are interconnected. Thus, in this paper, an accuracy analysis for a novel non-overlapping domain decomposition method with a variety of subdomain numbers which are interconnected at cross point is carried out. The cause of accuracy degradation is also analyze and establishment of countermeasure is discussed.

Finite Element Tearing and Interconnecting Method and its Algorithms for Parallel Solution of Magnetic Field Problems

Abstract Because of the exponential increase of computational resource requirement for numerical field simulations of more and more complex physical phenomena and more and more complex large problems in science and engineering practice, parallel processing appears to be an essential tool to handle the resulting large-scale numerical problems. One way of parallelization of sequential (singleprocessor) finite element simulations is the use of domain decomposition methods. Domain decomposition methods (DDMs) for parallel solution of linear systems of equations are based on the partitioning of the analyzed domain into sub-domains which are calculated in parallel while doing appropriate data exchange between those sub-domains. In this case, the non-overlapping domain decomposition method is the Lagrange multiplier based Finite Element Tearing and Interconnecting (FETI) method. This paper describes one direct solver and two parallel solution algorithms of FETI method. Finally, comparative numerical tests demonstrate the differences in the parallel running performance of the solvers of FETI method. We use a single-phase transformer and a three-phase induction motor as twodimensional static magnetic field test problems to compare the solvers

High scalable non-overlapping domain decomposition method using a direct method for finite element analysis

In this paper, a novel domain decomposition method is introduced to solve a large finite element model. In this method, the decomposed domains do not overlap and are connected using a simple connective finite element, which influences the nodal point equilibrium between adjacent finite elements. This approach has the advantage that it allows use of a direct method such as Gauss elimination even in a singular problem. The singular stiffness matrices from the floating domain without the Dirichlet boundary conditions are changed into invertible stiffness matrices by assembling the connective elements. Another advantage is that computational time and storage can be reduced by using a banded matrix in the direct solver. In order to describe this proposed method, we first review the FETI method, which is the most popular domain decomposition method. Then the proposed method is introduced with a technical approach in a distributed computer system. Finally, the high scalability and computational efficiency of the proposed method are verified by comparing with the traditional FETI method for 2D and 3D finite element examples which have floating subdomains. In the result, we demonstrate high scalability for a large finite element model.

Parallel Nonoverlapping DDM Combined with the Characteristic Method for Incompressible Miscible Displacements in Porous Media

Two types of approximation schemes are established for incompressible miscible displacements in porous media. First, standard mixed finite element method is used to approximate the velocity and pressure. And then parallel non-overlapping domain decomposition methods combined with the characteristics method are presented for the concentration. These methods use the characteristic method to handle the material derivative term of the concentration equation in the subdomains and explicit flux calculations on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the velocity and pressure can be approximated simultaneously, and the parallelism can be achieved for the concentration equation. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability. These schemes hold the advantages of nonoverlapping domain decomposition methods and the characteristic method. Optimal error estimates in L2-norm are derived for these two schemes, respectively.

Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation

Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method for the mixed neutron diffusion equations, discretized with Raviart–Thomas–Nédélec finite elements. This method is based on the Schwarz iterative algorithm with Robin interface conditions to handle communications. We analyse this method from the continuous point of view to the discrete point of view, and we give some numerical results in a realistic highly heterogeneous 3D configuration. Computations are carried out with the MINOS solver of the APOLLO3®1APOLLO3 is a registered trademark in France.1 neutronics code.

Domain Decomposition for Stokes-Darcy Flows with Curved Interfaces

A non-overlapping domain decomposition method is developed for coupled Stokes-Darcy flows in irregular domains. The Stokes region is discretized by standard Stokes finite elements while the Darcy region is discretized by the multipoint flux mixed finite element method. The subdomain grids may not match on the interfaces and mortar finite elements are employed to impose weakly interface continuity conditions. The interfaces can be curved and matching conditions are imposed via appropriate mappings from physical grids to reference grids with flat interfaces. The global problem is reduced to a mortar interface problem, which is solved by the conjugate gradient method. Each iteration involves solving subdomain problems of either Stokes or Darcy type, which is done in parallel. Computational experiments are presented to illustrate the convergence of the discretization and the condition number of the interface operator.

DOMAIN DECOMPOSITION FE-BI-MLFMA METHOD FOR SCATTERING BY 3D INHOMOGENEOUS OBJECTS

The hybrid flnite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) is a powerful method for calculating scattering by inhomogeneous objects. However, the conventional FE-BI-MLFMA often sufiers from iterative convergence problems. A non-overlapping domain decomposition method (DDM) is applied to FE-BI-MLFMA to speed up the iterative convergence. Furthermore, a preconditioner based on absorbing boundary condition and symmetric successive over relaxation (ABC-SSOR) is constructed to further accelerate convergence of the DDM-FE-BI-MLFMA. Numerical experiments demonstrate the e-ciency of the proposed preconditioned DDM-FE-BI-MLFMA.

Frequency-domain Elastic wave Simulation Based on the Nonoverlapping Domain Decomposition Method

A new wave simulation technique for the elastic wave equation in the frequency domain based on a no overlapping domain decomposition algorithm is investigated. The boundary conditions and the finite difference discrimination of the elastic wave equation are derived. The algorithm of no overlapping domain decomposition method is given. The method solves the elastic wave equation by iteratively solving sub problems defined on smaller sub domains. Numerical computations both for homogeneous and inhomogeneous media show the effectiveness of the proposed method. This method can be used in the full-waveform inversion.

A Nonoverlapping Domain Decomposition Method for an Exterior Anisotropic Quasilinear Elliptic Problem in Elongated Domains

Based on the Kirchhoff transformation, a nonoverlapping domain decomposition method is discussed for solving exterior anisotropic quasilinear problems with elliptic artificial boundary. By the principle of the natural boundary reduction, we obtain the natural integral equation for the anisotropic quasilinear problem on elliptic artificial boundaries and construct the algorithm and analyze its convergence. Moreover, we give the existence and uniqueness result for the original problem. Finally, some numerical examples are presented to illustrate the feasibility of the method.

A Multiscale Mortar Mixed Space Based on Homogenization for Heterogeneous Elliptic Problems

We consider a second order elliptic problem with a heterogeneous coefficient written in mixed form. The nonoverlapping mortar domain decomposition method is efficient in parallel if the mortar interface coupling space has a restricted number of degrees of freedom. In the heterogeneous case, we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom. In the case of a locally periodic heterogeneous coefficient of period $\varepsilon$, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when $\varepsilon$ is small. Moreover, we present three numerical examples to assess its performance when the coefficient is not obviously locally periodic. We show that the new mortar method works well, and better than polynomial mortar spaces.

Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations

This paper is concerned with global-in-time, nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem. Two approaches are considered: one uses the time-dependent Steklov--Poincaré operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interfaces between subdomains is derived, and different time grids are employed to adapt to different time scales in the subdomains. Demonstrations of the well-posedness of the Robin subdomain problems involved in the OSWR method and a convergence proof of the OSWR algorithm are given for the mixed formulation. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods.

A Nonoverlapping Domain Decomposition Method for Incompressible Stokes Equations with Continuous Pressures

A nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semidefinite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains fo...

Domain Decomposition Methods for Solving Stokes--Darcy Problems with Boundary Integrals

We consider a coupled problem of Stokes and Darcy equations. This involves solving PDEs of different orders simultaneously. To overcome this difficulty, we apply a nonoverlapping domain decomposition method based on a Robin boundary condition obtained by combining the velocity and force interface conditions. The coupled system is then reduced to solving each problem separately by an iterative procedure using a Krylov subspace method. The numerical solution in each subdomain is based on the boundary integral formulation, where the kernels are regularized and correction terms are added to reduce the regularization error.

Adjoint Based A Posteriori Analysis of Multiscale Mortar Discretizations with Multinumerics

In this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method. The error estimates are based on the solution of an appropriately defined adjoint problem. We present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain. The primal subdomains are discretized using either an interior penalty discontinuous Galerkin method or a continuous Galerkin method with weakly imposed Dirichlet conditions. The mixed subdomains are discretized using Raviart--Thomas mixed finite elements. The a posteriori error estimate also accounts for the errors due to adjoint-inconsistent subdomain discretizations. The coupling between the subdomain discretizations is achieved via a mortar space. We show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive adaptive refinement.