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16
A CLASS OF DISCONTINUOUS PETROVGALERKIN METHODS. PART I: THE TRANSPORT EQUATION
"... Abstract. Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailor ..."
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Cited by 36 (11 self)
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Abstract. Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size h and polynomial degree p, and outperforms the standard upwind DG method. 1.
A DISCRETE DUALITY FINITE VOLUME APPROACH TO HODGE DECOMPOSITION AND DIVCURL PROBLEMS ON ALMOST ARBITRARY TWODIMENSIONAL MESHES
"... We define discrete dierential operators such as grad, div and curl, on general twodimensional nonorthogonal meshes. These discrete operators verify discrete analogues of usual continuous theorems: discrete Green formulae, discrete Hodge decomposition of vector fields, vector curls have a vanishing ..."
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Cited by 15 (5 self)
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We define discrete dierential operators such as grad, div and curl, on general twodimensional nonorthogonal meshes. These discrete operators verify discrete analogues of usual continuous theorems: discrete Green formulae, discrete Hodge decomposition of vector fields, vector curls have a vanishing divergence and gradients have a vanishing curl. We apply these ideas to discretize divcurl systems. We give error estimates based on the reformulation of these systems into equivalent equations for the potentials. Numerical results illustrate the use of the method on several types of meshes, among which degenerating triangular meshes and nonconforming locally refined meshes.
APPROXIMATION OF THE EIGENVALUE PROBLEM FOR TIME HARMONIC MAXWELL SYSTEM BY CONTINUOUS LAGRANGE FINITE ELEMENTS
"... Abstract. We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1conforming finite elements. The key idea consists of controlling the divergence of the electric field in a fractional Sobolev space H−α with α ∈ ( 1, 1). The method is shown to be convergent and ..."
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Cited by 11 (1 self)
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Abstract. We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1conforming finite elements. The key idea consists of controlling the divergence of the electric field in a fractional Sobolev space H−α with α ∈ ( 1, 1). The method is shown to be convergent and spectrally 2 correct. 1.
THE APPROXIMATION OF THE MAXWELL EIGENVALUE PROBLEM USING A LEASTSQUARES METHOD
, 2005
"... In this paper we consider an approximation to the Maxwell’s eigenvalue problem based on a very weak formulation of two divcurl systems with complementary boundary conditions. We formulate each of these divcurl systems as a general variational problem with different test and trial spaces, i.e., th ..."
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Cited by 10 (2 self)
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In this paper we consider an approximation to the Maxwell’s eigenvalue problem based on a very weak formulation of two divcurl systems with complementary boundary conditions. We formulate each of these divcurl systems as a general variational problem with different test and trial spaces, i.e., the solution space is Ä 2 (Ω) ≡ (L 2 (Ω)) 3 and components in the test spaces are in subspaces of H 1 (Ω), the Sobolev space of order one on the computational domain Ω. A finiteelement leastsquares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the divcurl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given.
Complements on Hilbert Spaces and Saddle Point Systems
 Journal of Computational and Applied Mathematics, Volume 225, Issue
"... For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility LadyshenskayaBabušcaBrezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators onl ..."
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Cited by 8 (8 self)
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For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility LadyshenskayaBabušcaBrezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In the light of the new spectral results for the Schur complements, we review the classical BabušcaBrezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges provided that the inexact process for inverting the residual at each step has the relative error smaller than 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions. Key words: inexact Uzawa algorithms, saddle point system, multilevel methods, adaptive methods 1
A LEASTSQUARES APPROXIMATION METHOD FOR THE TIMEHARMONIC MAXWELL EQUATIONS
"... Abstract. In this paper we introduce and analyze a new approach for the numerical approximation of Maxwell's equations in the frequency domain. Our method belongs to the recently proposed family of negativenorm leastsquares algorithms for electromagnetic problems which have already been appl ..."
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Cited by 6 (0 self)
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Abstract. In this paper we introduce and analyze a new approach for the numerical approximation of Maxwell's equations in the frequency domain. Our method belongs to the recently proposed family of negativenorm leastsquares algorithms for electromagnetic problems which have already been applied to the electrostatic and magnetostatic problems as well as the Maxwell eigenvalue problem (see [5, 4]). The scheme is based on a natural weak variational formulation and does not employ potentials or \gauge conditions". The discretization involves only simple, piecewise polynomial, nite element spaces, avoiding the use of the complicated Nedelec elements. An interesting feature of this approach is that it leads to simultaneous approximation of the magnetic and electric elds, in contrast to other methods where one of the unknowns is eliminated and is later computed by dierentiation. More importantly, the resulting discrete linear system is wellconditioned, symmetric and positive denite. We demonstrate that the overall numerical algorithm can be eciently implemented and has an optimal convergence rate, even for problems with low regularity. 1.
Multilevel discretization of Symmetric Saddle Point Systems without the discrete LBB Condition
"... Using an inexact Uzawa algorithm at the continuous level, we study the convergence of multilevel algorithms for solving saddlepoint problems. The discrete stability LadyshenskayaBabušcaBrezzi (LBB) condition does not have to be satisfied. The algorithms are based on the existence of a multilevel ..."
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Cited by 3 (3 self)
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Using an inexact Uzawa algorithm at the continuous level, we study the convergence of multilevel algorithms for solving saddlepoint problems. The discrete stability LadyshenskayaBabušcaBrezzi (LBB) condition does not have to be satisfied. The algorithms are based on the existence of a multilevel sequence of nested approximation spaces for the constrained variable. The main idea is to maintain an accurate representation of the residual associated with the main equation at each step of the inexact Uzawa algorithm at the continuous level. The residual representation is approximated by a Galerkin projection. Whenever a sufficient condition for the accuracy of the representation fails to be satisfied, the representation of the residual is projected on the next (larger) space available in the prescribed multilevel sequence. Numerical results supporting the efficiency of the algorithms are presented for the Stokes equations and a div − curl system. Key words: inexact Uzawa algorithms, saddle point system, multilevel methods, adaptive methods
A NOTE ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF FREQUENCY DOMAIN ELASTIC WAVE PROBLEMS: A PRIORI ESTIMATES IN H 1.
"... Abstract. In this note, we provide existence and uniqueness results for frequency domain elastic wave problems. These problems are posed on the complement of a bounded domain (the scatterer). The boundary condition at infinity is given by the KupradzeSommerfeld radiation condition and involves diff ..."
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Cited by 3 (1 self)
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Abstract. In this note, we provide existence and uniqueness results for frequency domain elastic wave problems. These problems are posed on the complement of a bounded domain (the scatterer). The boundary condition at infinity is given by the KupradzeSommerfeld radiation condition and involves different Sommerfeld conditions on different components of the field. Our results are obtained by setting up the problem as a variational problem in the Sobolev space H 1 on a bounded domain. We use a nonlocal boundary condition which is related to the Dirichlet to Neumann conditions used for acoustic and electromagnetic scattering problems. We obtain stability results for the source problem, a necessary ingredient for the analysis of numerical methods for this problem based on finite elements or finite differences. 1.
MULTILEVEL GRADIENT UZAWA ALGORITHMS FOR SYMMETRIC SADDLE POINT PROBLEMS
"... Abstract. In this paper, we introduce a general multilevel gradient Uzawa algorithm for symmetric saddle point systems. We compare its performance with the performance of the standard Uzawa multilevel algorithm. The main idea of the approach is to combine a double inexact Uzawa algorithm at the con ..."
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Cited by 2 (2 self)
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Abstract. In this paper, we introduce a general multilevel gradient Uzawa algorithm for symmetric saddle point systems. We compare its performance with the performance of the standard Uzawa multilevel algorithm. The main idea of the approach is to combine a double inexact Uzawa algorithm at the continuous level with a gradient type algorithm at the discrete level. The algorithm is based on the existence of a priori multilevel sequences of nested approximation pairs of spaces, but the family does not have to be stable. To ensure convergence, the process has to maintain an accurate representation of the residuals at each step of the inexact Uzawa algorithm at the continuous level. The residual representations at each step are approximated by projections or representation operators. Sufficient conditions for ending the iteration on a current pair of discrete spaces are determined by computing simple indicators that involve consecutive iterations. When compared with the standard Uzawa multilevel algorithm, our proposed algorithm has the advantages of automatically selecting the relaxation parameter, lowering the number of iterations on each level, and improving on running time. By carefully choosing the discrete spaces and the projection operators, the error for the second component of the solution can be significantly improved even when comparison is made with the discretization on standard families of stable pairs. 1.
Cascadic multilevel algorithms for symmetric saddle point systems
 Comput. Math. Appl
"... Abstract. In this paper, we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive ..."
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Cited by 2 (2 self)
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Abstract. In this paper, we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems. On each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a Schur complement is implemented. The level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error. We prove new estimates that relate the iteration error and the residual for the constraint equation. The new estimates are the key ingredients in imposing an efficient level change criterion. The first iteration on each new level uses information about the best approximation of the discrete solution from the previous level. The theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a nonincreasing number of iterations on each level. Even though numerical results supporting the efficiency of the algorithms are presented for the Stokes system, the algorithms can be applied to a large class of boundary value problems, including first order systems that can be reformulated at the continuous level as symmetric saddle point problems, such as the Maxwell equations. 1.