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Unified analysis of discontinuous Galerkin methods for elliptic problems
 SIAM J. Numer. Anal
, 2001
"... Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment ..."
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Cited by 519 (31 self)
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Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 320 (25 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 225 (39 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
An overview of projection methods for incompressible flows
 Comput. Methods Appl. Mech. Engrg
"... Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1no ..."
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Cited by 203 (20 self)
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Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1norm for the velocity and of the L 2norm for the pressure. We also show that the class of fractionalstep methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75–111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414–443] can be interpreted as the rotational form of our velocitycorrection methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L 2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods. Key words. Navier–Stokes equations, projection methods, fractionalstep methods, incompressibility, finite elements, spectral approximations
Finite element exterior calculus, homological techniques, and applications
 ACTA NUMERICA
, 2006
"... ..."
Finite elements in computational electromagnetism
, 2002
"... This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinateindependent statement of Maxwell’s equations in the calculus of differential forms. Th ..."
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Cited by 137 (7 self)
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This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinateindependent statement of Maxwell’s equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete dif
Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows
 SIAM J. Sci. Comput
"... this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is ..."
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Cited by 120 (17 self)
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this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is devoted to the derivation of weighted a posteriori error bounds for linear functionals of the solution. Finally, in Section 4 we present some numerical examples to demonstrate the performance of the resulting adaptive finite element algorithm. 2 Model problem and discretisation Given an open bounded polyhedral domain fl in lI n, n _> 1, with boundary 0fl, we consider the following problem: find u: fl > lI m, m _> 1, such that div(u) = 0 in , (2.1) where, ,: m __> mxn is continuously differentiable. We assume that the system of conservation laws (2.1) may be supplemented by appropriate initial/boundary conditions. For example, assuming that B(u, y) := EiL1 biVu'(u) has m real eigenvalues and a complete set of linearly independent eigenvectors for all y = (yl,, Yn) C n; then at inflow/outflow boundaries, we require that B(u, n)(u g) = 0, where n denotes the unit outward normal vector to 0fl, B(u, n) is the negative part of B(u, n) and g is a (given) realvalued function. To formulate the discontinuous Galerkin finite element method (DGFEM, for short) for (2.1), we first introduce some notation. Let 7 = {n} be an admissible subdivision of fl into open element domains n; here h is a piecewise constant mesh function with h(x) = diam(n) 2 Houston e al. when x is in element n. For p Iq0, we define the following finite element space n,  {v [L()]": vl [%(n)] " Vn }, where Pp(n) denotes the set of polynomials of degree at most p over n. Given that v [Hi(n)] m for each n...
Preconditioning stochastic Galerkin saddle point systems
 SIAM J. Matrix Anal. Appl
"... Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDE ..."
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Cited by 110 (4 self)
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Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are blockdense and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steadystate diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of blockdiagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing socalled Kronecker product preconditioners we improve the robustness of cheap, meanbased preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.
deal.II – a general purpose object oriented finite element library
 ACM TRANS. MATH. SOFTW
"... An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced objectoriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be ..."
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Cited by 105 (28 self)
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An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced objectoriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be arranged to fit users requirements. Through this approach, deal.II supports a large number of different applications covering a wide range of scientific areas, programming methodologies, and applicationspecific algorithms, without imposing a rigid framework into which they have to fit. A judicious use of programming techniques allows to avoid the computational costs frequently associated with abstract objectoriented class libraries. The paper presents a detailed description of the abstractions chosen for defining geometric information of meshes and the handling of degrees of freedom associated with finite element spaces, as well as of linear algebra, input/output capabilities and of interfaces to other software, such as visualization tools. Finally, some results obtained with applications built atop deal.II are shown to demonstrate the powerful capabilities of this toolbox.
Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as CellCentered Finite Differences
 SIAM J. NUMER. ANAL
, 1997
"... We present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor ..."
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Cited by 97 (45 self)
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We present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order RaviartThomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cellcentered finite difference method, requiring the solution of a sparse, positive semideflnite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order ap proximations in the L a and HSnorms (and superconvergence is obtained between the Laprojection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If h denotes the maximal mesh spacing, then the optimal rate is O(h). The superconvergence rate O(h ) is obtained for the scalar unknown and rate O(h 3/) for its gradient and flux in certain discrete norms; moreover, the full O(h ) is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.