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1,077
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. Sci. Comput
, 2006
"... Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a no ..."
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Cited by 52 (22 self)
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Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schöberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2P0 and isoP2P1 finite elements in support of our conclusions. We also show results of a comparison with two stateoftheart preconditioners, showing the competitiveness of our approach. Key words. Navier–Stokes equations, finite element, iterative methods, multigrid, preconditioning AMS subject classifications. 65F10, 65N22, 65F50 DOI. 10.1137/050646421 1. Introduction. We consider the numerical solution of the steady Navier– Stokes equations governing the flow of a Newtonian, incompressible viscous fluid. Let Ω ⊂ R d (d =2,3) be a bounded, connected domain with a piecewise smooth
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 51 (6 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...
An A Priori Error Analysis Of The Local Discontinuous Galerkin Method For Elliptic Problems
, 2000
"... . In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L 2 norm of the gradient and ..."
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Cited by 51 (19 self)
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. In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L 2 norm of the gradient and the L 2 norm of the potential are of order k and k + 1=2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h \Gamma1 are taken, the order of convergence of the potential increases to k + 1. The optimality of these theoretical results are tested in a series of numerical experiments on two dimensional domains. Key words. Finite elements, discontinuous Galerkin methods, elliptic problems AMS subject classifications. 65N30 1. Introduction. In this paper, we present the first a priori error analysis of the Local Discontinuous Galerkin (LDG) method for the following classical model elliptic problem: \Gamma\Deltau = f in\Omega ; u ...
Mixed finite element methods on nonmatching multiblock grids
 SIAM J. Numer. Anal
, 2000
"... Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux ..."
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Cited by 50 (26 self)
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Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux condition. A standard mixed finite element method is used within the blocks. Optimal order convergence is shown for both the solution and its flux. Moreover, at certain discrete points, superconvergence is obtained for the solution and also for the flux in special cases. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory.
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 49 (17 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.
Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes
 SIAM J. Numer. Anal
, 2007
"... The stability and convergence properties of the mimetic finite difference method for diffusiontype problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved. 1 ..."
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Cited by 48 (10 self)
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The stability and convergence properties of the mimetic finite difference method for diffusiontype problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved. 1
Adaptive Wavelet Methods II  Beyond the Elliptic Case
 FOUND. COMPUT. MATH
, 2000
"... This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmet ..."
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Cited by 47 (14 self)
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This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [DKS]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now wellposed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a signicant modication of the ideas from [CDD]. The main departure from [CDD] is to develop an iterative scheme that directly applies to the innite dimensional problem rather than nite subproblems derived from the infinite problem. This rests on an adaptive application of the innite dimensional operator to finite vectors representing elements from finite dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding waveletbest Nterm approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces like the LBB condition no longer arise.
A Posteriori Finite Element Bounds for LinearFunctional Outputs of Elliptic Partial Differential Equations
 Computer Methods in Applied Mechanics and Engineering
, 1997
"... We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is base ..."
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Cited by 46 (9 self)
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We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic "energy" reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine "truthmesh" discretization are then derived by appealing to a dual maxmin relaxation evaluated for optimally chosen adjoint and hybridflux candidate Lagrange multipliers generated by a Kelement coarser "workingmesh" approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomainlocal, symmetric Neumann pro...
A Multigrid Algorithm For The Mortar Finite Element Method
 SIAM J. NUMER. ANAL
"... The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar finite element discretization of second order elliptic boundary value problems. In order to establish the infsup condition for the saddle point formulation and to motivate ..."
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Cited by 45 (11 self)
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The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar finite element discretization of second order elliptic boundary value problems. In order to establish the infsup condition for the saddle point formulation and to motivate the subsequent treatment of the discretizations we revisit first briefly the theoretical concept of the mortar finite element method. Employing suitable meshdependent norms we verify the validity of the LBB condition for the resulting mixed method and prove an L 2 error estimate. This is the key for establishing a suitable approximation property for our multigrid convergence proof via a duality argument. In fact, we are able to verify optimal multigrid efficiency based on a smoother which is applied to the whole coupled system of equations. We conclude with several numerical tests of the proposed scheme which confirm the theoretical results and show the efficiency and the robustness of the method even in situations not covered by the theory.
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
 MATH. COMP
, 2002
"... The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly h ..."
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Cited by 44 (10 self)
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The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an oversampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random lognormal relative permeability to demonstrate the efficiency and accuracy of the proposed method.