Robin Transmission Conditions Research Articles

Optimized Schwarz waveform relaxation method for the incompressible Stokes problem

We propose and analyse the optimized Schwarz waveform relaxation (OSWR) method for the unsteady incompressible Stokes equations. Well-posedness of the local subdomain problems with Robin boundary conditions is proved. Convergence of the velocity is shown through energy estimates; however, pressure converges only up to constant values in the subdomains, and an astute correction technique is proposed to recover these constants from the velocity. The convergence factor of the OSWR algorithm is obtained through a Fourier analysis, and allows to efficiently optimize the space-time Robin transmission conditions involved in the OSWR method. Then, numerical illustrations for the two-dimensional unsteady incompressible Stokes system are presented to illustrate the performance of the OSWR algorithm.

Analyzing electromagnetic scattering from complex multi-layer patch objects using a multi-trace domain decomposition method

The local-coupling multi-trace and domain decomposition method originally developed to analyze electromagnetic scattering from closed composite objects is extended to solve the scattering from complex multilayer patch objects. Different from the traditional global radiation coupling surface integral equation domain decomposition method and contact-region modeling method (CRM), the local-coupling property yields many excellent properties, highly sparse matrix, better computational performance, etc. In this novel proposed method, the multilayer patch objects are decomposed into many independent sub-domains, and each sub-domain is assumed as a homogeneous dielectric subdomain to formulate the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE) for dielectrics as the governing equations. The corresponding boundary condition of the perfect electric conducting (PEC) surface sheet is then achieved by adopting Robin transmission conditions (RTCs) to PEC surfaces, termed PEC-RTCs. The final resulting equation yield a better performance than the traditional method. Numerical examples validate the advantages of the proposed method.

Optimized Schwarz Methods for the Cahn–Hilliard Equation

The Cahn–Hilliard equation was originally proposed to describe the phase separation phenomenon for a binary alloy in the quenching process and now has been widely applied in many other scientific fields. To investigate its solution behavior, one mainly depends on the numerical simulations because of the nonlinearity. However, it would require very small time steps to describe the phase separation procedure since it evolves very quickly in time, which implies that an efficient solver for the spatial problem at each time level is very important. In this paper, we investigate a coupled second order elliptic system of constant coefficients, which occurs in many different unconditionally energy stable time discretization schemes for the Cahn–Hilliard equation. For a min-max problem of convergence factor that is obtained via Fourier analysis in the case of two-subdomain domain decomposition, by using an asymptotic analysis we obtain the optimized transmission parameters in explicit form applied in the Robin and the two-sided Robin transmission conditions for both the overlapping and the nonoverlapping Schwarz domain decomposition algorithms, and we obtain as well the corresponding asymptotic convergence rates. Particularly, we also explore and analyze the many-subdomain cases directly and find that the algorithms are scalable until a certain number of subdomains is reached for small time steps, even if there is no coarse grid correction. We finally use numerical examples to illustrate our theoretical findings.

A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing

In this work, a wave-equation-based discontinuous Galerkin (DG) method hybridized with the Robin transmission condition (DG-RTC) is developed to solve the frequency-domain electromagnetic (EM) problems. The proposed DG method directly discretizes the vector electric field wave equation in each subdomain, and subsequently, a term named numerical flux is introduced at the subdomain interfaces to connect the solutions between neighboring subdomains. However, the numerical flux depends not only on the electric field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> but also on the magnetic field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> residing over the interface. Thereby, another equation is essential for solving <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> simultaneously. Realizing that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> only situates at the interface of adjacent subdomains, it is thus desired that the auxiliary equation also merely involves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> at the interfaces. To reach this aim, at the subdomain interfaces, a Robin transmission condition (RTC) based on the tangential continuity of EM fields across the interface is introduced as the second equation. With this proposed DG-RTC algorithm, only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {E}$ </tex-math></inline-formula> is a volume variable, while <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}$ </tex-math></inline-formula> is a two-dimensional (2-D) one, and the degrees of freedom (DoFs) are thus greatly reduced compared with the traditional DG method. On the other hand, in the frequency domain, the established subdomain matrix equations are implicitly coupled with each other, resulting in a global matrix system. Directly solving it is not yet cheap. To alleviate the computational cost, a finite-element tearing and interconnecting (FETI)-like approach resorts. With the block-diagonal preconditioner and a restriction operator, the global matrix is decomposed into a surface matrix equation and several local matrix equations pertinent to each subdomain. In this way, a direct solver can be applied to solve these small matrix equations efficiently. To validate the proposed DG method in solving various EM problems, several representative examples are presented.

Reconstruction of extended regions in EIT with a generalized Robin transmission condition

Reconstruction of extended regions in EIT with a generalized Robin transmission condition

Reconstruction of small and extended regions in EIT with a Robin transmission condition

We consider an inverse shape problem coming from electrical impedance tomography with a Robin transmission condition. In general, a boundary condition of Robin type models corrosion. In this paper, we study two methods for recovering an interior corroded region from electrostatic data. We consider the case where we have small volume and extended regions. For the case where the region has small volume, we will derive an asymptotic expansion of the current gap operator and prove that a MUSIC-type algorithm can be used to recover the region. In the case where one has an extended region, we will show that the regularized factorization method can be used to recover said region. Numerical examples will be presented for both cases in two dimensions in the unit circle.

A Multitrace Surface Integral Equation Solver to Simulate Graphene-Based Devices

A multitrace surface integral equation (MT-SIE) solver is proposed to analyze electromagnetic field interactions on composite devices involving magnetized and nonmagnetized graphene sheets. The computation domain is decomposed into two subdomains: an exterior subdomain that represents the unbounded background medium where the device resides in an interior subdomain that represents the dielectric substrate. Resistive Robin transmission conditions (RRTCs) are formulated to describe the infinitesimally thin graphene sheet that partially covers the surface between the two subdomains. On the rest of this surface, traditional Robin transmission conditions (RTCs) are enforced. The electric and magnetic field equations are used as the governing equations in each subdomain. The governing equations of a subdomain are locally coupled to the governing equations of its neighbor using RRTCs and RTCs. The accuracy and the applicability of the proposed MT-SIE solver are demonstrated by various numerical examples.

A Hybridizable Discontinuous Galerkin Time-Domain Method With Robin Transmission Condition for Transient Thermal Analysis of 3-D Integrated Circuits

In this article, a discontinuous Galerkin time-domain (DGTD) method hybridized with Robin transmission condition (RTC-DGTD) is proposed for the transient thermal analysis of 3-D integrated circuits (ICs). As a kind of domain decomposition method (DDM), the DGTD method employs a term named numerical flux for the information exchange of solutions between neighboring subdomains. The numerical flux is a function of both temperature <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and heat flux q, and thus for the heat equation, apart from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> , the variable q has to be solved as well. To reach this, the traditional DGTD method decomposes the second-order partial-differential thermal equation into two first-order ones, which results that both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and q need to be solved in the whole computational domain, thereby the computational cost is extremely expensive. In this work, to get this trouble-treated, the proposed DGTD method is directly applied to discretize the second-order heat equation. An RTC is established at the interfaces of subdomains, which serves as the second governing equation and meanwhile bridges another connection of solutions in adjacent subdomains. In this way, the new variable q is now only present at the interfaces of subdomains. Consequently, the total number of unknowns is significantly reduced compared with that in the traditional DGTD method. To further speed up the time-marching scheme, an adaptive time-stepping method in terms of the proportional–integral–derivative (PID) control algorithm is employed. Besides, by introducing a restriction operator, the globally coupled matrix equation is torn into a number of small matrix equations pertinent to each subdomain, which is later solved via a finite-element tearing and interconnecting (FETI)-like method. To validate and verify the accuracy and efficiency of the proposed algorithm, several representative examples are studied and compared with COMSOL simulations.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

DC IR-Drop Analysis of Power Distribution Networks by a Robin Transmission Condition-Enhanced Discontinuous Galerkin Method

In this work, a novel Robin transmission condition (RTC)-enhanced discontinuous Galerkin (DG) method is proposed for the dc IR-drop analysis of power distribution networks with Joule heating effects included. Unlike the conventional DG method, the proposed DG method straightforwardly applied to discretize the second-order spatial partial differential governing equations for the electrostatic potential <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> and the steady-state temperature <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> . The numerical flux in DG used to facilitate the information exchange among neighboring subdomains introduces two additional variables: the current density <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$J$ </tex-math></inline-formula> for the electrical potential equation and the thermal flux <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> for the thermal equation. To solve them, at the interface of neighboring subdomains, an RTC is presented as the second equation to establish another connection for solutions in neighboring subdomains. With this strategy, the number of degrees of freedom (DoFs) involved in the proposed RTC-DG method is dramatically reduced compared with the traditional DG method. The finalized matrix system is solved in an finite-element tearing and interconnecting (FETI)-like procedure, namely the unknowns are obtained in a subdomain-by-subdomain scheme. Finally, the accuracy and the efficiency of the proposed RTC-DG method are validated by several representative examples.

A Flexible FEM-BEM-DDM for EM Scattering by Multiscale Anisotropic Objects

In this article, a nonconformal domain decomposition method (DDM) based on the hybrid finite element method (FEM) and boundary element method (BEM) is proposed to solve arbitrary 3-D anisotropic (including bianisotropic and single anisotropic) multiscale composite objects. The object is divided into interior and exterior regions. The interior region is solved by the bianisotropic finite element domain decomposition method (FEM-DDM), and the exterior region is solved by the discontinuous Galerkin (DG) method based on the combined field integral equation (CFIE). In order to decouple the electric and magnetic fields of the bianisotropic object, a new finite element–boundary element–domain decomposition method (FEM-BEM-DDM) formulation is derived. To ensure the continuity of electric current and electric field between anisotropic FEM subdomains, a new auxiliary current is introduced into the interface of different subdomains of FEM and the interface of FEM subdomains with BEM subdomains. The former uses second-order transmission conditions (SOTCs) to ensure continuity, whereas the latter uses Robin transmission conditions (RTCs) to ensure continuity. The BEM dense matrix–vector multiplication is accelerated by a multilevel fast multipole algorithm (MLFMA). The proposed FEM-BEM-DDM allows for nonconformal discretization between any touching subdomains and provides an effective preconditioner. In numerical examples, the accuracy and effectiveness of the method are verified.

Nonconforming time discretization based on Robin transmission conditions for the Stokes–Darcy system

We consider a space-time domain decomposition method based on Schwarz waveform relaxation (SWR) for the time-dependent Stokes–Darcy system. The coupled system is formulated as a time-dependent interface problem based on Robin–Robin transmission conditions, for which the decoupling SWR algorithm is proposed and proved for the convergence. In this approach, the Stokes and Darcy problems are solved independently and globally in time, thus allowing the use of different time steps for the local problems. Numerical tests are presented for both non-physical and physical problems with various mesh sizes and time step sizes to illustrate the accuracy and efficiency of the proposed method.

Discrete Optimization of Robin Transmission Conditions for Anisotropic Diffusion with Discrete Duality Finite Volume Methods

Discrete Duality Finite Volume (DDFV) methods are very well suited to discretize anisotropic diffusion problems, even on meshes with low mesh quality. Their performance stems from an accurate reconstruction of the gradients between mesh cell boundaries, which comes however at the cost of using both a primal (cell centered) and a dual (vertex centered) mesh, and thus leads to larger system sizes. To solve these systems, we propose to use non-overlapping optimized Schwarz methods with Robin transmission conditions, which can also well take into account anisotropic diffusion across subdomain interfaces. We study these methods here directly at the discrete level, and prove convergence using energy estimates for general decompositions including cross points and fully anisotropic diffusion. Our analysis reveals that primal and dual meshes might be coupled using different optimized Robin parameters in the optimized Schwarz methods. We present both the separate and coupled optimization of Robin transmission conditions and derive parameters which lead to the fastest possible convergence in each case. We illustrate our results with numerical experiments for the model problem, and also in situations that go beyond our analysis, with an application to anisotropic image reconstruction.

A Multitrace Surface Integral Equation Method for PEC/Dielectric Composite Objects

The multitrace domain decomposition surface integral equation (MT-DD-SIE) originally developed to analyze electromagnetic scattering from dielectric composite objects is extended to efficiently account for perfect electrically conducting (PEC) bodies. This is achieved by adopting Robin transmission conditions (RTCs) to PEC surfaces. These PEC-RTCs, which are the only governing equations for a PEC body, are used to “couple” it to the dielectric bodies. Upon discretization, PEC-RTCs produce a well-conditioned matrix block and, therefore, does not negatively affect the convergence of the iterative solution of the MT-DD-SIE matrix equation. The resulting method is significantly faster and has a smaller memory footprint than the traditional globally coupled contact-region-modeling method that uses the coupled electric field and Poggio-Miller-Chang-Harrington-Wu-Tsai SIEs in analyzing electromagnetic scattering from composite PEC/dielectric objects. This is demonstrated by numerical examples involving electrically large scatterers.

Convergence Analysis of Schwarz Waveform Relaxation for Nonlocal Diffusion Problems

Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions applied to the one-dimensional and two-dimensional Helmholtz and Maxwell's equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm being applied to the equations with absorption, then, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that are typically encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments.

A Local Coupling Multitrace Domain Decomposition Method for Electromagnetic Scattering From Multilayered Dielectric Objects

In this article, a local coupling multitrace domain decomposition method (LCMT-DDM) based on surface integral equation (SIE) formulations is proposed to analyze electromagnetic scattering from multilayered dielectric objects. Different from the traditional SIE-DDM, where the interactions between subdomains are accounted for using global radiation coupling, LCMT-DDM uses a local coupling scheme. The original multilayered object is decomposed into several independent domains, i.e., the exterior region (free space) and many homogeneous interior regions (dielectrics). The boundaries of subdomains are all touching faces, where only the Robin transmission conditions (RTCs) are enforced to ensure the field continuity. Hence, each subdomain only couples with its neighboring regions, which makes the DDM system a highly sparse matrix, especially when the number of subdomains is large. In each subdomain, the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE) for dielectrics are used as the governing equations. By imposing RTCs, well-conditioned equations are formed in each subdomain without invoking the combined-field integral equation (CFIE), which usually causes accuracy issues in dielectric modeling. Since the subdomain matrices are diagonally dominant, the flexible generalized minimal residual (FGMRES) technique is used to accelerate the iterative solution of the whole DDM system. Moreover, an effective preconditioner that can be recursively constructed is proposed.

Schwarz Solvers and Preconditioners for the Closest Point Method

The discretization of surface intrinsic elliptic partial differential equations (PDEs) poses interesting challenges not seen in flat spaces. The discretization of these PDEs typically proceeds by either parametrizing the surface, triangulating the surface, or embedding the surface in a higher dimensional flat space. The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in the embedding space to their closest points on the surface. In the CPM, this mapping also serves as an extension operator that brings surface intrinsic data onto the embedding space, allowing PDEs to be numerically approximated by standard methods in a narrow tubular neighborhood of the surface. We focus on numerically approximating the positive Helmholtz equation, $\left(c-\Delta_{\mathcal{S}}\right)u=f,~c\in\mathbb{R}^+$, by the CPM paired with finite differences. This yields a large, sparse, and nonsymmetric system to be solved. Herein, we develop restricted additive Schwarz (RAS) and optimized restricted additive Schwarz (ORAS) solvers and preconditioners for this discrete system. In particular, we develop a general strategy for computing overlapping partitions of the computational domain and defining the corresponding Dirichlet and Robin transmission conditions. We demonstrate that the convergence of the ORAS solvers and preconditioners can be improved by using a modified transmission condition where more than two overlapping subdomains meet. Numerical experiments are provided for a variety of analytical and triangulated surfaces. We find that ORAS solvers and preconditioners outperform their RAS counterparts and that, as expected using domain decomposition (DD) as a preconditioner rather than as a solver gives faster convergence. The methods exhibit good parallel scalability over the range of process counts tested.

Explicit Computation of Robin Parameters in Optimized Schwarz Waveform Relaxation Methods for Schrödinger Equations Based on Pseudodifferential Operators

The Optimized Schwarz Waveform Relaxation algorithm, a domain decomposition method based on Robin transmission condition, is becoming a popular computational method for solving evolution partial differential equations in parallel. Along with well-posedness, it offers a good balance between convergence rate, computational complexity and simplicity of the implementation. The fundamental question is the selection of the Robin parameter to optimize the convergence of the algorithm. In this paper, we propose an approach to explicitly estimate the Robin parameter which is based on the approximation of the transmission operators at the subdomain interfaces, for the linear/nonlinear SchrodingerSchr¨Schrodinger equation. Some illustrating numerical experiments are proposed for the one-and two-dimensional problems.

Twofold Domain Decomposition Method for the Analysis of Multiscale Composite Structures

A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.