A non-conforming domain decomposition for Raviart-Thomas vector fields in three dimensions

In this paper we are concerned with a domain decomposition method with non-matching grids for Raviart-Thomas finite elements. In this method, the normal complement of the resulting approximation is not continuous across the interface. To handle such non-conformity, a new matching condition will be introduced. Such matching condition still results in a symmetric and positive definite stiffness matrix. It will be shown that the approximate solution generated by the domain decomposition possesses the optimal energy error estimate.     

A regularized domain decomposition method with Lagrange multiplier

In this paper, we are concerned with the nonoverlapping domain decomposition method with Lagrange multiplier for three-dimensional second-order elliptic problems with no zeroth-order term. It is known that the methods result in a singular subproblem on each internal (floating) subdomain. To handle the singularity, we propose a regularization technique which transforms the corresponding singular problems into approximate positive definite problems. For the regularized method, one can build the interface equation of the multiplier directly. We first derive an optimal error estimate of the regularized approximation, and then develop a cheap preconditioned iterative method for solving the interface equation. For the new method, the cost of computation will not be increased comparing the case without any floating subdomain. The effectiveness of the new method will be confirmed by both theoretical analyzes and numerical experiments. The work is supported by Natural Science Foundation of China G10371129.     

A mortar edge element method with nearly optimal convergence for three-dimensional Maxwell's equations

In this paper, we are concerned with mortar edge element methods for solving three-dimensional Maxwell's equations. A new type of Lagrange multiplier space is introduced to impose the weak continuity of the tangential components of the edge element solutions across the interfaces between neighboring subdomains. The mortar edge element method is shown to have nearly optimal convergence under some natural regularity assumptions when nested triangulations are assumed on the interfaces. A generalized edge element interpolation is introduced which plays a crucial role in establishing the nearly optimal convergence. The theoretically predicted convergence is confirmed by numerical experiments.

     

Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann–Neumann methods for hpfinite-element approximations of scalar elliptic problems ongeometrically refined boundary layer meshes in three dimensions.These are meshes that are highly anisotropic where the aspectratio typically grows exponentially with the polynomial degree.The condition number of our preconditioners is shown to be independentof the aspect ratio of the mesh and of potentially large jumpsof the coefficients. In addition, it only grows polylogarithmicallywith the polynomial degree, as in the case of p approximationson shape-regular meshes. This work generalizes our previousone on two-dimensional problems in Toselli & Vasseur (2003a,submitted to Numerische Mathematik, 2003c to appear in Comput.Methods Appl. Mech. Engng.) and the estimates derived here canbe employed to prove condition number bounds for certain typesof FETI methods.     

Substructuring preconditioners for the three fields domain decomposition method

We study a class of preconditioners based on substructuring, for the discrete Steklov-Poincaré operator arising in the three fields formulation of domain decomposition in two dimensions. Under extremely general assumptions on the discretization spaces involved, an upper bound is provided on the condition number of the preconditioned system, which is shown to grow at most as ( and denoting, respectively, the diameter and the discretization mesh-size of the subdomains). Extensive numerical tests--performed on both a plain and a stabilized version of the method--confirm the optimality of such bound.

     

Multigrid methods for H(div) in three dimensions with nonoverlapping domain decomposition smoothers

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.     

A domain decomposition and mixed method for a linear parabolic boundary value problem

In this paper, we discuss the convergence of a domain decompositionmethod for the solution of linear parabolic equations in theirmixed formulations. The subdomain meshes need not be quasi-uniform;they are composed of triangles or quadrilaterals that do notmatch at interfaces. For the ease of computation, this lackof continuity is compensated by a mortar technique based onpiecewise constant (discontinuous) multipliers. It is shownthat the method on triangles, parallelograms or slightly distortedparallelograms is convergent at the expense of a half-orderloss of accuracy compared with mortar methods based on piecewiselinear multipliers.     

A domain decomposition method on nested domains and nonmatching grids

A one directionally coupled problem on two nested domains is analyzed. The global domain and the subdomain are discretized by two triangulations that do not match on the subdomain. The connection between the two grids is established by using a stable projection operator onto the interface. An a priori error analysis is carried out and several numerical examples are given. The method is ideally suited for the case of a moving subdomain. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 374–387, 2004.     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

Substructuring preconditioners for saddle-point problems arising from Maxwell's equations in three dimensions

This paper is concerned with the saddle-point problems arising from edge element discretizations of Maxwell's equations in a general three dimensional nonconvex polyhedral domain. A new augmented technique is first introduced to transform the problems into equivalent augmented saddle-point systems so that they can be solved by some existing preconditioned iterative methods. Then some substructuring preconditioners are proposed, with very simple coarse solvers, for the augmented saddle-point systems. With the preconditioners, the condition numbers of the preconditioned systems are nearly optimal; namely, they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size.

     

A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems

A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.

     

Suitable iterative methods for solving the linear system arising in the three fields domain decomposition method

There are two approaches for applying substructuring preconditioner for the linear system corresponding to the discrete Steklov–Poincaré operator arising in the three fields domain decomposition method for elliptic problems. One of them is to apply the preconditioner in a common way, i.e. using an iterative method such as preconditioned conjugate gradient method [S. Bertoluzza, Substructuring preconditioners for the three fields domain decomposition method, I.A.N.-C.N.R, 2000] and the other one is to apply iterative methods like for instance bi-conjugate gradient method, conjugate gradient square and etc. which are efficient for nonsymmetric systems (the preconditioned system will be nonsymmetric). In this paper, second approach will be followed and extensive numerical tests will be presented which imply that the considered iterative methods are efficient.     

A posteriori estimate and asymptotic partial domain decomposition

The method of asymptotic partial decomposition of a domain aims at replacing a 3D or 2D problem by a hybrid problem 3D???1D; or 2D???1D, where the dimension of the problem decreases in part of the domain. The location of the junction between the heterogeneous problems is asymptotically estimated in certain circumstances, but for numerical simulations it is important to be able to determine the location of the junction accurately. In this article, by reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. Some numerical results are presented for a toy problem.     

The modified method of characteristics with mixed finite element domain decomposition procedures for the transient behavior of a semiconductor device

For the transient behavior of a semiconductor device, the modified method of characteristics with mixed finite element domain decomposition procedures applicable to parallel arithmetic is put forward. The electric potential equation is described by the mixed finite element method, and the electric, hole concentration and heat conduction equations are treated by the modified method of characteristics finite element domain decomposition methods. Some techniques, such as calculus of variations, domain decomposition, characteristic method, energy method, negative norm estimate and prior estimates and techniques are employed. Optimal order estimates in L² norm are derived for the error in the approximation solution. Thus the well‐known theoretical problem has been thoroughly and completely solved.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 353–368 2012     

Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions

Lower bounds for the condition numbers of the preconditioned systems are obtained for the Bramble-Pasciak-Schatz substructuring preconditioner and the Neumann-Neumann preconditioner in two dimensions. They show that the known upper bounds are sharp.

     

On the convergence rate of a parallel nonoverlapping domain decomposition method

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(h1/2 H-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(h1/2H-1/2)in a certain case no matter how parameter is chosen.     

On the domain decomposition method for the generalized Stokes problem with continuous pressure

Using the nonoverlapping domain decomposition approach, we propose a formulation of the dual Schur algorithm for the generalized Stokes problem discretized by a mixed finite element method continuous for the pressure in each subdomain, but discontinuous at the interfaces. The corresponding LBB condition is checked. The dual interface problem is written in the case of two subdomains, and it is generalized to several subdomains. An efficient preconditioner for the interface problem is derived. Numerical results are presented for two different local solvers. Parallel computations were made on an IBM‐SP2. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 84–106, 2000     

An optimization‐based domain decomposition method for the Boussinesq equations

We study an optimization based domain decomposition method for the Boussinesq equations governing natural convection problems. Domain decomposition is cast into a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the interface between solid and fluid subdomains. We showthat solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined. Finite element approximations of the solutions of the optimality system are examined. The domain decomposition method is also reformulated as a nonlinear least‐squares problem and the results of some numerical experiments are given. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 1–25, 2002     

A domain decomposition algorithm for elliptic problems in three dimensions

Summary Most domain decomposition algorithms have been developed for problems in two dimensions. One reason for this is the difficulty in devising a satisfactory, easy-to-implement, robust method of providing global communication of information for problems in three dimensions. Several methods that work well in two dimension do not perform satisfactorily in three dimensions.A new iterative substructuring algorithm for three dimensions is proposed. It is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain. The condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains. The new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions.This work was supported in part by the National Science Foundation under grant NSF-CCR-8903003 and by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Automatic domain decomposition on unstructured grids (DOUG)

This paper describes a parallel iterative solver for finite element discretisations of elliptic partial differential equations on 2D and 3D domains using unstructured grids. The discretisation of the PDE is assumed to be given in the form of element stiffness matrices and the solver is automatic in the sense that it requires minimal additional information about the PDE and the geometry of the domain. The solver parallelises matrix–vector operations required by iterative methods and provides parallel additive Schwarz preconditioners. Parallelisation is implemented through MPI. The paper contains numerical experiments showing almost optimal speedup on unstructured mesh problems on a range of four platforms and in addition gives illustrations of the use of the package to investigate several questions of current interest in the analysis of Schwarz methods. The package is available in public domain from the home page http://www.maths.bath.ac.uk/^∼mjh/. This revised version was published online in August 2006 with corrections to the Cover Date.