Non-conformal Domain Decomposition Research Articles

A primal–dual active set strategy for non-linear multibody contact problems

Non-conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations. For the discretization of a non-linear multibody contact problem, we use the mortar approach with a dual Lagrange multiplier space. To handle the non-linearity of the contact conditions, we apply a primal–dual active set strategy to find the actual contact zone. The algorithm can be easily generalized to multibody contact problems. A suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case. Using an inexact primal–dual active set strategy in combination with a multigrid method yields an efficient iterative solver. Different numerical examples for one and multibody contact problems illustrate the performance of the method.

A domain decomposition method for domains with holes using a complementary decomposition

In this paper, a new non-conforming domain decomposition method based on fictitious domains is considered for linear elasticity problems on domains with holes. Existence and uniqueness of the solution of this alternative approach are showed in the continuous and discrete case. Although the established a priori estimate for the discretization error is not optimal, the scheme is of interest in case of moving geometries and in shape optimization. To solve the arising coupled linear system, we use a block SOR method. The convergence rate does not depend on the mesh size. Numerical results illustrate the flexibility of the discretization scheme.

Residual a posteriori error estimates for a 3D mortar finite-element method: the Stokes system

We consider a non-conforming domain decomposition technique for the discretization of the three-dimensional Stokes equations by the mortar finite-element method. Relying on the velocity-pressure formulation of the system, we perform the numerical analysis of residual error indicators for this problem and we prove that the error estimators provide upper and lower bounds for the energy norm of the mortar finite-element solution.

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 Stabilized Domain Decomposition Method with Nonmatching Grids for the Stokes Problem in Three Dimensions

We present and study a nonconforming domain decomposition method for the discretization of the three-dimensional Stokes problem in the velocity-pressure formulation. The approximation is based on some local mixed finite elements for nonmatching tetrahedral grids. The aim pursued is a systematic construction of the mortared discrete velocity space, the pressure being not subjected to any matching constraints across the interfaces. Using the bubble stabilization techniques, applied in Brezzi and Marini's paper to the three fields method [Math. Comp., 70 (2001), pp. 911--934], allows us to define an algorithm which is easy to implement. The numerical analysis relies on the pressure-splitting argument of Boland and Nicolaides and allows us to establish an inf-sup condition with a constant that does not depend on the mesh size or on the total number of the subdomains. Then, by the Berger--Scott--Strang lemma written down for our saddle point system we derive optimal accuracy results.

Monotone Multigrid Methods on Nonmatching Grids for Nonlinear Multibody Contact Problems

Nonconforming domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We use a generalized mortar method based on dual Lagrange multipliers for the discretization of a nonlinear contact problem between linear elastic bodies. In the case of unilateral contact problems, pointwise constraints occur and monotone multigrid methods yield efficient iterative solvers. Here, we generalize these techniques to nonmatching triangulations, where the constraints are realized in terms of weak integral conditions. The basic new idea is the construction of a nested sequence of nonconforming constrained spaces. We use suitable basis transformations and a multiplicative correction. In contrast to other approaches, no outer iteration scheme is required. The resulting monotone method is of optimal complexity and can be implemented as a multigrid method. Numerical results illustrate the performance of our approach in two and three dimensions.

Eddy-current calculations in three-dimensional moving structures

A nonconforming nonoverlapping domain decomposition method to approximate the eddy-current problem, formulated in terms of the modified magnetic vector potential, in three-dimensional moving structures, is presented. This approximation allows for nonmatching grids at the sliding interface and is based on the mortar element method combined with edge elements in space and finite differences in time. Numerical results illustrate how the method works and the influence of eddy currents on the field distribution.

A Slideing Mesh-Mortar Method for a two Dimensional Currents Model of Electric Engines

The paper concerns the application of a non-conforming domain decomposition method to the problem of the computation of induced currents in electric engines with moving conductors. The {\em eddy currents model} is considered as a quasi-static approximation of Maxwell equations and we study the two dimensional problem associated either to the modified magnetic vector potential or to the magnetic field. Two discretizations are proposed, the first based on curved finite elements and the second one on iso-parametric finite elements in both the static and moving parts. The coupling is obtain by means of the {\em mortar element method} and the approximation on the whole domain turns out to be non-conforming. In both cases optimal error estimates are provided. Numerical tests are then proposed in the case of standard first order finite elements to test the reliability and precision of the method. An application of the method to study a realistic eddy currents problem is also provided. TEL:: 0382-505652 KEYWORDS:: Eddy currents problem, nonconforming finite element approximation, domain decomposition methods EMAIL:: annalisa@dimat.unipv.it

Calculation of eddy currents with edge elements on non-matching grids in moving structures

In this paper we present a nonconforming nonoverlapping domain decomposition method to approximate the eddy current problem, formulated in terms of the electric field variable, in nonstationary structures. This approximation, that allows for nonmatching grids at the sliding interface, is based on the mortar element method combined with edge elements in space and finite differences in time. Numerical results illustrate how the method works and the influence of the free part movement on the electric field distribution.

Inf-sup conditions for the mortar spectral element discretization of the Stokes problem

Summary. The aim of this paper is to prove some Babuska–Brezzi type conditions which are involved in the mortar spectral element discretization of the Stokes problem, for several cases of nonconforming domain decompositions.

Preconditioners for nonconforming domain decomposition methods

In this article we briefly discuss two preconditioner techniques, the Neumann– Neumann-preconditioner and the Dirichlet–Neumann-preconditioner, for nonconforming domain decomposition methods. We will show with some examples that the numerical results obtained with nonconforming multidomain techniques are qualitatively equivalent to the results obtained on an unique domain.

Iterative Substructuring Preconditioners for Mortar Element Methods in Two Dimensions

The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal approximations. In this paper, we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric, second-order, elliptic equations in two dimensions. Both spectral and finite element methods, for geometrically conforming as well as nonconforming domain decompositions, are studied. In each case, we obtain a polylogarithmic bound on the condition number of the preconditioned matrix.

Conditioning of Pseudospectral Matrices for Certain Domain Decompositions

In this paper we examine the conditioning of the matrices resulting from certain conforming pseudospectral approximations. In particular, we consider non-conforming domain decompositions in rectangular domains and the solution of fourth order problems. We investigate the way in which the poor conditioning of these matrices affects the performance, in terms of accuracy, of various direct methods of solution of the resulting systems. We also show how a simple iterative refinement procedure can improve the accuracy of the results obtained with a capacitance technique.

Coupling Spectral and Finite Elements for Second Order Elliptic Three-Dimensional Equations

This paper deals with the approximation of elliptic second order problems with a nonconforming domain decomposition method that allows for coupling spectral and finite element discretizations in three dimensions. We present two kinds of matching conditions at the interfaces between the subdomains. The detailed numerical analysis of the error committed upon the exact solution turns out to be optimal. We describe also the implementation and give some numerical results confirming the theoretical predictions.

Direct methods for spectral approximations in nonconforming domain decompositions

The application of a conforming spectral collocation method to certain nonconforming domain decompositions leads to global matrices which have a particular block structure. We study the performance of various direct linear system solvers, some of which exploit this block structure, on a Cray J-916 vector computer, an SGI Power Challenge 8000, and an IBM RS6000 workstation.

La méthode des elements avec joint appliquée aux méthodes d'approximations Discrete Kirchhoff Triangles

This paper deals with the analysis of a non-conforming domain decomposition method, the mortar element method for a shell model approximated by a D.K.T. finite element method. We present a mathematical analysis of the method and derive optimal consistency error and best fit error.

Conforming spectral approximations for non-conforming domain decompositions

A spectral domain decomposition scheme is introduced for the numerical solution of second- and fourth-order elliptic problems. The technique is applicable to certain domain decompositions of rectangular or rectangularly decomposable domains. It is shown that it yields approximations which are pointwise C 0 continuous across the subdomain interfaces for second order problems and pointwise C 1 continuous across the subdomain interfaces for fourth-order problems.

Error estimators for mortar finite element discretizations

AbstractWe consider two different types of error estimators for a nonconforming domain decomposition method, the so‐called mortar finite element method. At the interfaces of the subdomains is no conformity condition enforced. An example is presented where we obtain strongly nonconforming triangulations within the adaptive refinement process.