A Rudin–Carleson theorem for planar vector fields

This paper generalizes the Rudin–Carleson theorem for homogeneous solutions of locally solvable real analytic vector fields.     

A continuation–domain decomposition algorithm for bifurcation problems

We study numerical solution branches of certain parameter-dependent problems defined on compact domains with various boundary conditions. The finite differences combined with the domain decomposition method are exploited to discretize the partial differential equations. We propose efficient numerical algorithms for solving the associated linear systems and for the detection of bifurcation points. Sample numerical results are reported. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Poincaré inequality for vector fields

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”. Both authors were partially supported by the University of Bologna, funds for selected research topics.     

A Dirichlet–Neumann reduced basis method for homogeneous domain decomposition problems

Reduced basis methods allow efficient model reduction of parametrized partial differential equations. In the current paper, we consider a reduced basis method based on an iterative Dirichlet–Neumann coupling for homogeneous domain decomposition of elliptic PDEʼs. We gain very small basis sizes by an efficient treatment of problems with a-priori known geometry. Moreover iterative schemes may offer advantages over other approaches in the context of parallelization. We prove convergence of the iterative reduced scheme, derive rigorous a-posteriori error bounds and provide a full offline/online decomposition. Different methods for basis generation are investigated, in particular a variant of the POD-Greedy procedure. Experiments confirm the rigor of the error estimators and identify beneficial basis construction procedures.     

Quasispaces induced by vector fields measurable in ℝ3

We study some metric functions that are induced by a class of basis vector fields in ?³ with measurable coordinates. These functions are proved to be quasimetrics in the domain of definition of the vector fields. Under some natural constraints, the Rashevsky-Chow Theorem and the Ball-Box Theorem are established for the classes of vector fields we consider.     

A spectral domain decomposition approach for the generalized Burger’s–Fisher equation

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Fisher equation (B–F). Firstly, theory of application of Chebyshev spectral collocation method (CSCM) and domain decomposition on the generalized Burger’s–Fisher equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.     

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.