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The Neumann–Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann–Neumann methods that have better convergence properties and require fewer computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov–Poincaré operators. [ABSTRACT FROM AUTHOR]

We prove convergence for the nonoverlapping Robin--Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov--Poincaré operators based on the p-structure and the Lp-generalization of the Lions--Magenes spaces. This framework allows the reformulation of the Robin--Robin method into a Peaceman--Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions. [ABSTRACT FROM AUTHOR]

A nonoverlapping domain decomposition technique applied to a finite difference method is presented for the numerical solution of the forward backward heat equation in the case of one-dimension. While the previous at tempts in dealing with this problem have been based on an iterative domain decomposition scheme, the current work avoids iterations. Also a physical matching condition is suggested to avoid difficulties caused by the interface boundary nodes. Furthermore, we obtain a square system of equations. In addition, the convergence and stability of the proposed method are investi gated. Some numerical experiments are given to show the effectiveness of the proposed method.

In this work, we consider compressible single-phase flow problems in a porous medium containing a fracture. In the fracture, a nonlinear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a nonlinear interface problem. We then introduce two new algorithms that are able to efficiently handle the nonlinearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve for the nonlinearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by precomputing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann codimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings X and in particular, t. The stability and the convergence of both schemes are obtained, where user-given parameters are optimized to minimize the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory. [ABSTRACT FROM AUTHOR]

This paper presents a 2D simulation of transient heat transfer in the human eye using appropriate boundary conditions. The mathematical model governing bioheat transfer in the human eye is discussed and the existence and uniqueness of the solution are proven. Four methods based on finite element method and nonoverlapping domain decomposition method to obtain transient heat transfer in the human eye are presented and described in details. After conducting numerous simulations using realistic parameters obtained from the open literature and after comparison with measurements reported by previous experimental studies, all proposed methods gave an accurate representation of transient heat transfer in the human eye. The results obtained by the domain decomposition of the human eye into four subdomains are found to be the closest to reality. [ABSTRACT FROM AUTHOR]

The paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model time-dependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a space-time interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of time-dependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, a bioheat model of temperature distribution in the human eye is studied, the mathematical formulation of this model is described using adequate mathematical tools. The existence and the uniqueness of the solution of this problem is proven and four algorithms based on finite element method approximation and domain decomposition methods are presented in details. The validation of all algorithm is done using a numerical application for an example where the analytical solution is known. The properties and parameters reported in the open literature for the human eye are used to approximate numerically the temperature for bioheat model by finite element approximation and nonoverlapping domain decomposition method. The obtained results that are verified using the experimental results recorded in the literature revealed a better accuracy by the use of algorithm proposed. [ABSTRACT FROM AUTHOR]

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

We consider optimal control problems for gas flow in pipeline networks. The equations of motion are taken to be represented by a first-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a network and introduce a tailored time discretization thereof. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a nonoverlapping domain decomposition of the optimal control problem on the graph into local problems on smaller sub-graphs--ultimately on single edges. We prove convergence of the domain decomposition method on networks and study the wellposedness of the corresponding time-discrete optimal control problems. The point of the paper is that we establish virtual control problems on the decomposed subgraphs such that the corresponding optimality systems are in fact equal to the systems obtained via the domain decomposition of the entire optimality system. [ABSTRACT FROM AUTHOR]

Macro-hybrid mixed variational models of two-phase flow, through fractured porous media, are analyzed at the mesoscopic and macroscopic levels. The mesoscopic models are treated in terms of nonoverlapping domain decompositions, in such a manner that the porous rock matrix system and the fracture network interact across rock-rock, rock-fracture, and fracture-fracture interfaces, with flux transmission conditions dualized. Alternatively, the models are scaled to a macroscopic level via an asymptotic process, where the width of the fractures tends to zero, and the fracture network turns out to be an interface system of one less spatial dimension, with variable high permeability. The two-phase flow is characterized by a fractional flow dual mixed variational model. Augmented two-field and three-field variational reformulations are presented for regularization, internal approximations, and macro-hybrid mixed finite element implementation. Also abstract proximal-point penalty-duality algorithms are derived and analyzed for parallel computing. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the plane wave methods for Helmholtz equations with large wave numbers. We extend the plane wave weighted least-squares (PWLS) method and the plane wave discontinuous Galerkin (PWDG) method to Helmholtz equations in inhomogeneous mediums and with mixed boundary conditions. The main goal of this paper is to construct parallel preconditioners for solving the resulting discrete systems. To this end, we design a kind of substructuring domain decomposition preconditioner for both the two-dimensional case and the three-dimensional case, which is different from the existing substructuring preconditioners. We apply the proposed preconditioner to solve the Helmholtz systems generated by the PWLS method or the PWDG method, and we find that the new preconditioners with practical coarse mesh sizes possess relatively stable convergence, i.e., the iteration counts of the corresponding iterative methods (PCG or preconditioned GMRES) increase slowly when the wave number increases (and the mesh size decreases). [ABSTRACT FROM AUTHOR]

In the present paper we propose a new iterative method for the numerical solution of system of linear algebraic equations with symmetric positive definite and positive semidefinite matrices arising from the approximation of diffusion equations by the mixed hybrid finite element method on triangular and tetrahedral meshes. The method is based on a combination of the block steepest descent idea and nonoverlapping domain decomposition algorithm. We give the formulation of the method and a simple convergence estimate in the case of regular shaped quasiuniform meshes. [ABSTRACT FROM AUTHOR]

We consider a fully implicit formulation for two-phase flow in a porous medium with capillarity, gravity, and compressibility in three dimensions. The method is implicit in time and uses the multiscale mortar mixed finite element method for a spatial discretization in a nonoverlapping domain decomposition context. The interface conditions between subdomains are enforced in terms of Lagrange multiplier variables defined on a mortar space. The novel approach in this work is to linearize the coupled system of subdomain and mortar variables simultaneously to form a global Jacobian. This algorithm is shown to be more efficient and robust compared to previous algorithms that relied on two separate nested linearizations of subdomain and interface variables. We also examine various upwinding methods for accurate integration of phase mobility terms near subdomain interfaces. Numerical tests illustrate the computational benefits of this scheme. [ABSTRACT FROM AUTHOR]

SUMMARY Starting from a recently developed energetic space-time weak formulation of boundary integral equations for wave propagation problems, a coupling algorithm is presented, which allows a flexible use of FEMs and BEMs as local discretization techniques. Emphasis is given to theoretical and experimental analysis of the stability of the proposed method. Several numerical results on model problems are presented and discussed, showing that both bounded and unbounded three-dimensional domains can be efficiently addressed. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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 ε, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when e 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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we demonstrate a local convergence of an adaptive scalar solver which is practical for strongly diagonal dominant Jacobian problems such as in some systems of nonlinear equations arising from the application of a nonoverlapping domain decomposition method. The method is tested to a nonlinear interface problem of a multichip heat conduction problem. The numerical results show that the method performs slightly better than a Newton–Krylov method. [Copyright &y& Elsevier]

Nonoverlapping domain decomposition methods have been shown to provide efficient iterative algorithms for the solution of the time-harmonic Maxwell's equations. Convergence of the algorithms depends strongly upon the nature of the transmission conditions that communicate information between adjacent subdomains. In this work, we introduce a new second order transmission condition that improves convergence. In contrast to previous high order interface conditions, the new condition uses two second order transverse derivatives to improve the convergence of evanescent modes without deteriorating the convergence of propagating modes. An analysis using a splitting of the field into traverse electric and magnetic components demonstrates the improved convergence provided by the transmission condition. Numerical experiments demonstrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, optimized Schwarz methods with Robin transmission conditions are considered for solving the time-dependent linear algebraic systems resulting from finite element approximations for parabolic problems. We use both analysis and numerical experiments to investigate the convergence behavior. Particularly, we check the influence of the time step size on the convergence rate. We get the estimate of the upper bound of convergence rate 1 - O(min{h1/2H-1/2, h1/2H3/2t-1}), where h is the mesh size, H is the size of subdomains, and t is the time step size. Our numerical results show that the convergence rate of this method is fast. We also search for the optimal parameter λ by experiments and find the rule of its distribution. [ABSTRACT FROM AUTHOR]

In recent years, a nonoverlapping domain decomposition iterative procedure, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed. It is known that the convergence rate of this method is 1 − O( h), where h is mesh size. In this paper, the convergence rate is improved to be 1 − O( h 1/2H−1/2) sometime by choosing suitable parameter, where H is the subdomain size. Counter examples are constructed to show that our convergence estimates are sharp, which means that the convergence rate cannot be better than 1 − O( h 1/2H−1/2) in a certain case no matter how parameter is chosen. [ABSTRACT FROM AUTHOR]

A nonoverlapping domain decomposition approach with uniform and matching grids is used to define and compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on a square partitioned into four squares. The collocation solution on four interfaces is computed using the preconditioned conjugate gradient method with the preconditioner defined in terms of interface preconditioners for the adjacent squares. The collocation solution on four squares is computed by a matrix decomposition method that uses fast Fourier transforms. With the number of preconditioned conjugate gradient iterations proportional to log2 N, the total cost of the algorithm is O( N2 log2 N), where the number of unknowns in the collocation solution is O( N2). The approach presented in this paper, along with that in [B.Bialecki and M.Dryja, A nonoverlapping domain decomposition method for orthogonal spline collocation problems. SIAM J. Numer. Anal. (2003) 41, 1709 – 1728.], generalizes to variable coefficient equations on rectangular polygons partitioned into many subrectangles and is well suited for parallel computation. [ABSTRACT FROM AUTHOR]

Abstract: Two different interface predictions are considered, in this work, for the solution of multi-dimensional parabolic problem defined over nonoverlapping subdomains. These interface boundary conditions are predicted by the explicit prediction (EP) and the time lagging (TL) methods with δx and 2δx mesh spacings away from the interface lines. We considered the additive splitting up method with respect to the spatial variable to solve the multi-dimensional parabolic problem over each subdomains. For more accurate solution we present the correction for the explicit interface prediction and implement the proposed methods for the solution of two different model problems. [Copyright &y& Elsevier]

In recent years, a nonoverlapping Robin-type domain decomposition method (DDM) for the finite element discretization systems of the second order elliptic equations, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed since it was first proposed by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in Proceedings of the 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1990, pp. 202-223]. However, the convergence rate of this DDM with many subdomains remains open when the lower term of equations vanishes. This open problem will be considered in this paper. The convergence rate is almost 1 — O(h¹/²H-1/²) in certain cases—for example, the case of a small number of subdomains, where h is the mesh size and H is the size of subdomain. In order to get the desirous convergence results, two mathematics skills are introduced in this paper; one is complexification of real linear space and the other is the spectral radius formula. [ABSTRACT FROM AUTHOR]

A class of corrected explicit-implicit domain decomposition (CEIDD) methods is investigated for the parallel approximation of linear heat equations. Explicit-implicit domain decomposition (EIDD) methods are computationally and communicationally efficient for each time step but always suffer from small time step size restrictions. By adding an interface correction step to Kuznetsov's EIDD, the one-dimensional CEIDD procedure achieves unconditional stability without discarding the time-stepwise efficiency of the EIDD method. In order to maintain the virtues of the CEIDD method and improve the flexibility in domain partitioning, for solving multidimensional problems, special zigzag-shaped interfaces are suggested in the CEIDD method. Based on non-crossover and crossover types of zigzag interfaces, the resulting CEIDD-ZI algorithms are studied for two strategies of subdomain partition. By the energy method, it shows that the proposed algorithms, including their degenerate cases—the corrected explicit hopscotch schemes-are convergent in the discrete H¹ seminorm and L² norm. Numerical experiments confirm the results in our analysis. [ABSTRACT FROM AUTHOR]

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed reaction–diffusion problem. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process. The rate of convergence of the monotone Schwarz method is estimated. Numerical experiments are presented. [Copyright &y& Elsevier]

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping sub domains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

A nonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on an L-shaped region. We prove existence and uniqueness of the collocation solution and derive optimal order Hs-norm error bounds for s = 0, 1, 2. The collocation solution on two interfaces is computed using the preconditioned conjugate gradient method, and the collocation solution on three squares is computed by a matrix decomposition method that uses fast Fourier transforms. The total cost of the algorithm is O(N² log N), where the number of unknowns in the collocation solution is O(N²). [ABSTRACT FROM AUTHOR]