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34
Multigrid method for H(DIV) in three dimensions
 ETNA
, 1997
"... . We are concerned with the design and analysis of a multigrid algorithm for H(div; ## elliptic linear variational problems. The discretization is based on H(div; conforming RaviartThomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector ..."
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Cited by 32 (4 self)
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. We are concerned with the design and analysis of a multigrid algorithm for H(div; ## elliptic linear variational problems. The discretization is based on H(div; conforming RaviartThomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector fields in the kernel of the divergence operator and its complement is paramount. We exploit the representation of discrete solenoidal vector fields as curls of finite element functions in socalled Nedelec spaces. It turns out that a combined nodal multilevel decomposition of both the RaviartThomas and Nedelec finite element spaces provides the foundation for a viable multigrid method. Its GauSeidel smoother involves an extra stage where solenoidal error components are tackled. By means of elaborate duality techniques we can show the asymptotic optimality in the case of uniform refinement. Numerical experiments confirm that the typical multigrid efficiency is actually achieved for model...
A POSTERIORI ERROR ESTIMATES FOR MAXWELL EQUATIONS
"... Maxwell equations are posed as variational boundary value problems in the function space H(curl) and are discretized by Nédélec finite elements. In [4], a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove th ..."
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Cited by 32 (3 self)
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Maxwell equations are posed as variational boundary value problems in the function space H(curl) and are discretized by Nédélec finite elements. In [4], a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasiinterpolation operators introduced recently in [22]. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.
OPTIMIZED SCHWARZ METHODS FOR MAXWELL’S EQUATIONS
, 2009
"... Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is i ..."
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Cited by 29 (16 self)
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Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell’s equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell’s equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments.
An Iterative Substructuring Method For RaviartThomas Vector Fields In Three Dimensions
 SIAM J. Numer. Anal
, 2000
"... . The iterative substructuring methods, also known as Schur complement methods, form one of two important families of domain decomposition algorithms. They are based on a partitioning of a given region, on which the partial differential equation is defined, into nonoverlapping substructures. The pr ..."
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Cited by 21 (4 self)
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. The iterative substructuring methods, also known as Schur complement methods, form one of two important families of domain decomposition algorithms. They are based on a partitioning of a given region, on which the partial differential equation is defined, into nonoverlapping substructures. The preconditioners of these conjugate gradient methods are then defined in terms of local problems defined on individual substructures and pairs of substructures, and, in addition, a global problem of low dimension. An iterative method of this kind is introduced for the lowest order RaviartThomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure. The theoretical bounds are confirmed by a series of numerical experiments. Key words. RaviartThomas finite elements, domain decomposition, ...
An iterative substructuring method for Maxwell's equations in two dimensions
 Math. Comp
, 1998
"... Abstract. Iterative substructuring methods, also known as Schur complement methods, form an important family of domain decomposition algorithms. They are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner ..."
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Cited by 19 (4 self)
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Abstract. Iterative substructuring methods, also known as Schur complement methods, form an important family of domain decomposition algorithms. They are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner. For conforming finite element approximations of H 1,itisknownthat the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures, and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure. In this paper, the same result is established for similar iterative methods for loworder Nédélec finite elements, which approximate H(curl; Ω) in two dimensions. Results of numerical experiments are also provided. 1.
Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations
 Math. Comp
, 2003
"... Abstract. Time harmonic Maxwell equations in lossless media lead to a second order differential equation for electric field involving a differential operator that is neither elliptic nor definite. A Galerkin method using Nedelec spaces can be employed to get approximate solutions numerically. The pr ..."
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Cited by 17 (7 self)
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Abstract. Time harmonic Maxwell equations in lossless media lead to a second order differential equation for electric field involving a differential operator that is neither elliptic nor definite. A Galerkin method using Nedelec spaces can be employed to get approximate solutions numerically. The problem of preconditioning the indefinite matrix arising from this method is discussed here. Specifically, two overlapping Schwarz methods will be shown to yield uniform preconditioners. 1.
A new approximation technique for divcurl systems
 MATH. COMP
, 2003
"... In this paper, we describe an approximation technique for divcurl systems based in (L 2 (Ω) 3) where Ω is a domain in R 3. We formulate this problem as a general variational problem with different test and trial spaces. The analysis requires the verification of an appropriate infsup condition. This ..."
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Cited by 16 (4 self)
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In this paper, we describe an approximation technique for divcurl systems based in (L 2 (Ω) 3) where Ω is a domain in R 3. We formulate this problem as a general variational problem with different test and trial spaces. The analysis requires the verification of an appropriate infsup condition. This results in a very weak formulation where the solution space is (L 2 (Ω)) 3 and the data reside in various negative norm spaces. Subsequently, we consider finite element approximations based on this weak formulation. The main approach of this paper involves the development of “stable pairs ” of discrete test and trial spaces. With this approach, we enlarge the test space so that the discrete infsup condition holds and we use a negativenorm leastsquares formulation to reduce to a uniquely solvable linear system. This leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low Sobolev regularity (e.g., (H s (Ω)) 3 with 0 < s < 1/2). The resulting algebraic equations are symmetric, positive definite and well conditioned. A second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned. Some numerical results are also presented.
A FETI Domain Decomposition Method For Maxwell's Equations With Discontinuous Coefficients In Two Dimensions
 Courant Institute, New York University
, 1999
"... . A class of FETI methods for the edge element approximation of vector eld problems in two dimensions is introduced and analyzed. First, an abstract framework is presented for the analysis of a class of FETI methods where a natural coarse problem, associated with the substructures, is lacking. Then, ..."
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Cited by 12 (5 self)
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. A class of FETI methods for the edge element approximation of vector eld problems in two dimensions is introduced and analyzed. First, an abstract framework is presented for the analysis of a class of FETI methods where a natural coarse problem, associated with the substructures, is lacking. Then, a family of FETI methods for edge element approximations is proposed. It is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures. The estimate is also independent of the jumps of both of the coecients of the original problem. Numerical results validating the theoretical bounds are given. The method and its analysis can be easily generalized to Raviart{Thomas element approximations in two and three dimensions. Key words. Edge elements, Maxwell's equations, domain decomposition, FETI, preconditioners, heterogeneous coecients AMS subjec...
Overlapping Schwarz methods in H(curl) on polyhedral domains
 Journal of Numerical Mathematics
"... Abstract. We consider domain decomposition preconditioners for the linear algebraic equations which result from finite element discretization of problems involving the bilinear form α(·, ·) + (curl ·, curl ·) defined on a polyhedral domain Ω. Here (·, ·) denotes the inner product in (L2(Ω))3 and α i ..."
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Cited by 12 (1 self)
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Abstract. We consider domain decomposition preconditioners for the linear algebraic equations which result from finite element discretization of problems involving the bilinear form α(·, ·) + (curl ·, curl ·) defined on a polyhedral domain Ω. Here (·, ·) denotes the inner product in (L2(Ω))3 and α is a positive number. We use Nedelec’s curlconforming finite elements to discretize the problem. Both additive and multiplicative overlapping Schwarz preconditioners are studied. Our results are uniform with respect to the mesh size and α under standard assumptions concerning the overlapping subdomains. 1.
Substructuring preconditioners for saddlepoint problems arising from Maxwell’s equations in three dimensions
, 2002
"... Abstract. This paper is concerned with the saddlepoint 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 saddlepoint ..."
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Cited by 11 (8 self)
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Abstract. This paper is concerned with the saddlepoint 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 saddlepoint 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 saddlepoint 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.