Non-overlapping Subdomains Research Articles

A Mortar Mixed Finite Volume Method for Elliptic Problems on Non-matching Multi-block Triangular Grids

A mixed finite volume method is considered for the mixed formulation of second-order elliptic equations. The computational domain can be decomposed into non-overlapping sub-domains or blocks and the diffusion tensors may be discontinuous across the sub-domain boundaries. We define a conforming triangular partition on each sub-domain independently, and employ the standard mixed finite volume method within each sub-domain. A mortar finite element space is introduced to approximate the trace of the pressure on the non-matching interfaces. Moreover, a continuity condition of flux is imposed weakly. We prove the scheme’s first order optimal rate of convergence for both the pressure and the velocity. Numerical experiments are provided to illustrate the error behavior of the scheme and confirm our theoretical results.

Primal Domain Decomposition Methods for the Total Variation Minimization, Based on Dual Decomposition

We propose nonoverlapping domain decomposition methods for solving the total variation minimization problem. We decompose the domain of the dual problem into nonoverlapping rectangular subdomains, where local total variation problems are solved. We convert the local dual problems into the equivalent primal forms which reproduce the original problem at smaller dimensions. Sequential and parallel algorithms are presented. The convergence of both algorithms is analyzed and numerical results are presented.

A Novel JMCFIE-DDM for Analysis of EM Scattering and Radiation by Composite Objects

In this letter, a new integral equation domain decomposition method (IE-DDM) using electric and magnetic current combined-field integral equation (JMCFIE) is proposed for efficient electromagnetic analysis of composite objects. This method provides a flexible way to decompose a complex composite domain into several easily solvable nonoverlapping subdomains. In order to obtain a good approximation of the global solution, both electric and magnetic field interactions are considered at the subdomain interfaces, in addition to the traditional Robin-type transmission conditions. In this manner, the unphysical reflections at subdomain interfaces can be further suppressed. It is shown that the condition of the new JMCFIE-DDM system is much improved in this new implementation. Accuracy and efficiency of the proposed method are demonstrated through several numerical experiments.

The mass-preserving and modified-upwind splitting DDM scheme for time-dependent convection–diffusion equations

In the paper, a new mass-preserving and modified-upwind S-DDM scheme over non-overlapping multi-block sub-domains for solving time-dependent convection–diffusion equations in two dimensions is developed and analyzed. On each sub-domain, the intermediate fluxes on the interfaces of sub-domains are firstly computed by the modified semi-implicit flux schemes. Then, the solutions and fluxes in the interiors of sub-domains are computed by the modified-upwind splitting implicit solution and flux coupled schemes. On the theoretical aspect, analyzing the mass-preserving and modified-upwind S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique, the conservative modified-upwind technique, and the domain decomposition. We prove the proposed scheme to be mass conservative and unconditionally stable in discrete L2-norm and further prove its convergence and obtain error estimates. Numerical experiments are presented to illustrate mass conservation, convergence and parallelism.

A Multisolver Scheme Based on Robin Transmission Conditions for Electromagnetic Modeling of Highly Complex Objects

A multisolver scheme based on Robin transmission conditions (RTCs) is proposed for electromagnetic modeling of highly complex objects. Different from the traditional finite element–boundary integral (FE-BI) method that applies BI equations to truncate the FE domain, the proposed multisolver scheme employs both FE and BI equations to model an object along with its background. To be specific, the entire computational domain consisting of the object and its background is first decomposed into multiple nonoverlapping subdomains with each modeled by either an FE or BI equation. The equations in the subdomains are then coupled into a multisolver system by enforcing the RTC at the subdomain interfaces. Finally, the combined system is solved iteratively with the application of an extended preconditioner based on an absorbing boundary condition and the multilevel fast multipole algorithm. To obtain an accurate solution, both the Rao–Wilton–Glisson and the Buffa–Christiansen functions are employed as the testing functions to discretize the BI equations. This scheme is applied to a variety of benchmark problems and the scattering from an aircraft with a launched missile to demonstrate its accuracy, versatility, and capability. The proposed scheme is compared with the multisolver scheme based on combined field integral equations to illustrate the differences between the two schemes.

A scalable schur-complement fluids solver for heterogeneous compute platforms

We present a scalable parallel solver for the pressure Poisson equation in fluids simulation which can accommodate complex irregular domains in the order of a billion degrees of freedom, using a single server or workstation fitted with GPU or Many-Core accelerators. The design of our numerical technique is attuned to the subtleties of heterogeneous computing, and allows us to benefit from the high memory and compute bandwidth of GPU accelerators even for problems that are too large to fit entirely on GPU memory. This is achieved via algebraic formulations that adequately increase the density of the GPU-hosted computation as to hide the overhead of offloading from the CPU, in exchange for accelerated convergence. Our solver follows the principles of Domain Decomposition techniques, and is based on the Schur complement method for elliptic partial differential equations. A large uniform grid is partitioned in non-overlapping subdomains, and bandwidth-optimized (GPU or Many-Core) accelerator cards are used to efficiently and concurrently solve independent Poisson problems on each resulting subdomain. Our novel contributions are centered on the careful steps necessary to assemble an accurate global solver from these constituent blocks, while avoiding excessive communication or dense linear algebra. We ultimately produce a highly effective Conjugate Gradients preconditioner, and demonstrate scalable and accurate performance on high-resolution simulations of water and smoke flow.

Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

Probabilistic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell’s equations as used in the magnetotelluric method. The domain is split into non-overlapping sub-domains and the solution on the sub-domain boundaries is obtained by evaluating the stochastic form of the exact solution of Maxwell’s equations by a Monte-Carlo approach. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell’s equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function-based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.

IE-DDM with a novel multiple-grid p-FFT for analyzing multiscale structures in half space

We present an integral equation domain decomposition method accelerated by a novel multiple-grid precorrected fast Fourier transform (MG-p-FFT) for the efficient analysis of multiscale structures in a half space. Based on the philosophy of DDM, the original computational domain is partitioned into several non-overlapping sub-domains. By employing non-conformal discretizations to each domain boundaries, combined field integral equation with half-space dyadic Green’s function is proposed for each individual sub-domain. Subsequently, the MG-p-FFT with auxiliary Cartesian grids with different size, order, location, and spacing, is adopted in each sub-domain independently to account for the self-interactions. Here, the proposed MG-p-FFT scheme outperforms the existing single-grid p-FFT scheme for multiscale problems by reducing the computational time and memory consumption. The proposed method can also be viewed as an effective preconditioning scheme for multiscale problems in a half space. The validity and advantages of the proposed method are illustrated by several representative numerical examples.

Mortar formulation for a class of staggered discontinuous Galerkin methods

A mortar formulation is developed and analyzed for a class of staggered discontinuous Galerkin (SDG) methods applied to second order elliptic problems in two dimensions. The computational domain consists of nonoverlapping subdomains and a triangulation is provided for each subdomain, which need not conform across subdomain interfaces. This feature allows a more flexible design of discrete models for problems with complicated geometries, shocks, or singular points. A mortar matching condition is enforced on the solutions across the subdomain interfaces by introducing a Lagrange multiplier space. Moreover, optimal convergence rates in both L2 and discrete energy norms are proved. Numerical results are presented to show the performance of the method.

Conservative tightly-coupled simulations of stochastic multiscale systems

Multiphysics problems often involve components whose macroscopic dynamics is driven by microscopic random fluctuations. The fidelity of simulations of such systems depends on their ability to propagate these random fluctuations throughout a computational domain, including subdomains represented by deterministic solvers. When the constituent processes take place in nonoverlapping subdomains, system behavior can be modeled via a domain-decomposition approach that couples separate components at the interfaces between these subdomains. Its coupling algorithm has to maintain a stable and efficient numerical time integration even at high noise strength. We propose a conservative domain-decomposition algorithm in which tight coupling is achieved by employing either Picard's or Newton's iterative method. Coupled diffusion equations, one of which has a Gaussian white-noise source term, provide a computational testbed for analysis of these two coupling strategies. Fully-converged (“implicit”) coupling with Newton's method typically outperforms its Picard counterpart, especially at high noise levels. This is because the number of Newton iterations scales linearly with the amplitude of the Gaussian noise, while the number of Picard iterations can scale superlinearly. At large time intervals between two subsequent inter-solver communications, the solution error for single-iteration (“explicit”) Picard's coupling can be several orders of magnitude higher than that for implicit coupling. Increasing the explicit coupling's communication frequency reduces this difference, but the resulting increase in computational cost can make it less efficient than implicit coupling at similar levels of solution error, depending on the communication frequency of the latter and the noise strength. This trend carries over into higher dimensions, although at high noise strength explicit coupling may be the only computationally viable option.

An Additive Schwarz Method Type Theory for Lions's Algorithm and a Symmetrized Optimized Restricted Additive Schwarz Method

Optimized Schwarz methods (OSMs) are very popular methods which were introduced by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), T. F. Chan, R. Glowinski, J. Periaux, and O. Widlund, eds., SIAM, Philadelphia, 1990, pp. 202--223] for elliptic problems and by B. Despres in [C. R. Acad. Sci. Paris Ser. I Math., 311 (1990), pp. 313--316] for propagative wave phenomena. We give here a theory for Lions's algorithm that is the genuine counterpart of the theory developed over the years for the Schwarz algorithm. The first step is to introduce a symmetric variant of the optimized restricted additive Schwarz (ORAS) algorithm [A. St-Cyr, M. J. Gander, and S. J. Thomas, SIAM J. Sci. Comput., 29 (2007), pp. 2402--2425] that is suitable for the analysis of a two-level method. Then we build a coarse space for which the convergence rate of the two-level m...

The Mass-Preserving S-DDM Scheme for Two-Dimensional Parabolic Equations

AbstractIn the paper, we develop and analyze a new mass-preserving splitting domain decomposition method over multiple sub-domains for solving parabolic equations. The domain is divided into non-overlapping multi-bock sub-domains. On the interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit) flux scheme. The solutions and fluxes in the interiors of sub-domains are computed by the splitting one-dimensional implicit solution-flux coupled scheme. The important feature is that the proposed scheme is mass conservative over multiple non-overlapping sub-domains. Analyzing the mass-preserving S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique and the domain decomposition at each time step. We prove theoretically that our scheme satisfies conservation of mass over multi-block non-overlapping sub-domains and it is unconditionally stable. We further prove the convergence and obtain the error estimate in L2-norm. Numerical experiments confirm theoretical results.

Solving EM Scattering From Multiscale Coated Objects With Integral Equation Domain Decomposition Method

This letter presents a nonconformal nonoverlapping integral equation domain decomposition method with impedance boundary condition (IE-DDM-IBC) to simulate electromagnetic scattering from multiscale coated objects. The proposed IE-DDM-IBC decomposes a multiscale coated object into nonoverlapping subdomains so that each subdomain can be independently meshed according to its geometrical feature. Besides rendering flexible mesh generation, the proposed IE-DDM-IBC also facilitates the derivation of an effective preconditioner that significantly accelerates the convergence of iterative solvers.

Convergence analysis for a second order Schwarz method for nonoverlapping subdomains

A second order Schwarz method for elliptic partial differential equations is developed and analyzed at a continuous level, which can be parallel implemented with local communications. It uses a second order boundary condition in terms of Poincaré–Steklov operators as its transmission condition on the interfaces. The convergence of the algorithm in H1 norm is obtained by using energy estimates, and convergence results hold on arbitrary geometry of subdomains with no cross points.

Analysis of multipatch discontinuous Galerkin IgA approximations to elliptic boundary value problems

In this work, we study the approximation properties of multipatch dG-IgA methods, that apply the multipatch Isogeometric Analysis discretization concept and the discontinuous Galerkin technique on the interfaces between the patches, for solving linear diffusion problems with diffusion coefficients that may be discontinuous across the patch interfaces. The computational domain is divided into non-overlapping subdomains, called patches in IgA, where B-splines, or NURBS approximations spaces are constructed. The solution of the problem is approximated in every subdomain without imposing any matching grid conditions and without any continuity requirements for the discrete solution across the interfaces. Numerical fluxes with interior penalty jump terms are applied in order to treat the discontinuities of the discrete solution on the interfaces. We provide a rigorous a priori discretization error analysis for diffusion problems in two- and three-dimensional domains, where solutions patchwise belong to $$W^{l,p}$$Wl,p, with some $$l\ge 2$$l?2 and $$ p\in ({2d}/{(d+2(l-1))},2]$$p?(2d/(d+2(l-1)),2]. In any case, we show optimal convergence rates of the discretization with respect to the dG - norm.

Splash jet and slamming generated by a rotating flap

The hydrodynamic problem of slamming generated by a rotating flap, commonly known as Oyster in the wave energy sector, plunging into water, is analysed based on the incompressible velocity potential theory. The problem is solved through the boundary element method in the time domain. Two typical case studies are undertaken. One is the flap plunging into calm water and the other into an incoming wave. The splash jet formed during the flap plunging is included in the simulation. When the jet meets the main flow, it is treated through the domain decomposition method without taking account the secondary impact, which is similar to the mathematical method of Riemann’s second sheet in the complex plane. The problem is solved in each non-overlapping subdomain, and the velocity and pressure continuity condition is imposed on the interface of the subdomains. Detailed results for the flap plunging into water with different velocities or accelerations are provided. The gravity and wave effects are also investigated.

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method (CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

Efficient combination of a 3D Quasi-Newton inversion algorithm and a vector dual-primal finite element tearing and interconnecting method

A Quasi-Newton method for reconstructing the constitutive parameters of three-dimensional (3D) penetrable scatterers from scattered field measurements is presented. This method is adapted for handling large-scale electromagnetic problems while keeping the memory requirement and the time flexibility as low as possible. The forward scattering problem is solved by applying the finite-element tearing and interconnecting full-dual-primal (FETI-FDP2) method which shares the same spirit as the domain decomposition methods for finite element methods. The idea is to split the computational domain into smaller non-overlapping sub-domains in order to simultaneously solve local sub-problems. Various strategies are proposed in order to efficiently couple the inversion algorithm with the FETI-FDP2 method: a separation into permanent and non-permanent subdomains is performed, iterative solvers are favorized for resolving the interface problem and a marching-on-in-anything initial guess selection further accelerates the process. The computational burden is also reduced by applying the adjoint state vector methodology. Finally, the inversion algorithm is confronted to measurements extracted from the 3D Fresnel database.

A Hybrid Solvers Enhanced Integral Equation Domain Decomposition Method for Modeling of Electromagnetic Radiation

The hybrid solvers based on integral equation domain decomposition method (HS-DDM) are developed for modeling of electromagnetic radiation. Based on the philosophy of “divide and conquer,” the IE-DDM divides the original multiscale problem into many closed nonoverlapping subdomains. For adjacent subdomains, the Robin transmission conditions ensure the continuity of currents, so the meshes of different subdomains can be allowed to be nonconformal. It also allows different fast solvers to be used in different subdomains based on the property of different subdomains to reduce the time and memory consumption. Here, the multilevel fast multipole algorithm (MLFMA) and hierarchical (H-) matrices method are combined in the framework of IE-DDM to enhance the capability of IE-DDM and realize efficient solution of multiscale electromagnetic radiating problems. The MLFMA is used to capture propagating wave physics in large, smooth regions, while H-matrices are used to capture evanescent wave physics in small regions which are discretized with dense meshes. Numerical results demonstrate the validity of the HS-DDM.

A Spectral Multidomain Penalty Method Solver for the Simulation of the Velocity Attenuation in Hyporheic Flows

A hydrodynamic model, based on a spectral multidomain penalty method (SMPM), is proposed to study the dissipation of the velocity fluctuations at the hyporheic zone. The model is based on the one dimensional Navier Stokes equations, assuming incompressibility and a hydrostatic approximation. The spatial discretization of the governing equations is done via the SMPM that is a multidomain collocation approach based on discontinuous non-overlapping subdomains that are connected by a penalty term that ensures stability of the solution by imposing weak continuity at the subdomain interfaces. The temporal discretization of the equations is handled with a high-order fractional time step technique. A spectral filter is used to stabilize the solution when spurious oscillations appear in the solution. A sinusoidal pulse is imposed at the water/sediment interface as a boundary condition, and the flow dynamics was evaluated under different conditions of kinematic viscosity. A progressive damping of the velocity fluctuations was observed in all cases, until full dissipation is reached. At that point, the flow can be governed by Darcy's law.