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Analysis of a multigrid preconditioner for CrouzeixRaviart discretization of elliptic PDE with jump coefficient
 Numer. Linear Algebra Appl
"... Abstract. In this paper, we present a multigrid Vcycle preconditioner for the linear system arising from piecewise linear nonconforming CrouzeixRaviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. W ..."
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Abstract. In this paper, we present a multigrid Vcycle preconditioner for the linear system arising from piecewise linear nonconforming CrouzeixRaviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rate of the multigrid Vcycle algorithm will deteriorate rapidly due to large jumps in coefficient. However, the preconditioned system has only a fixed number of small eigenvalues, which are deteriorated due to the large jump in coefficient, and the effective condition number is bounded logarithmically with respect to the mesh size. As a result, the multigrid Vcycle preconditioned conjugate gradient algorithm converges nearly uniformly. Numerical tests show both robustness with respect to jumps in the coefficient and the mesh size. 1.
A unified analysis of balancing domain decomposition by constraints for discontinuous Galerkin discretizations
 SIAM J. Numer. Anal
"... Abstract. The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of C(1 + log(H/h))2 is obtained for the condition number of the preconditioned system where C is a constant independent of h or H or large jumps in ..."
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Abstract. The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of C(1 + log(H/h))2 is obtained for the condition number of the preconditioned system where C is a constant independent of h or H or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of elementwise “local ” bilinear forms. The elementwise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms. Additionally, this new perspective enables a connection to be drawn between the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm.
Algebraic Multigrid for Discontinuous Galerkin Discretizations
 NUMER. LINEAR ALGEBRA APPL.
"... We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections and the first coarse level space is the subspace spanned by conti ..."
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We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections and the first coarse level space is the subspace spanned by continuous linear basis functions. The linear system associated with this space is constructed algebraically using a Galerkin approach with the natural embedding as the prolongation operator. For the construction of the linear systems on the subsequent coarser levels nonsmoothed aggregation AMG techniques are used. In a series of numerical experiments we establish the efficiency and robustness of the proposed method for various symmetric and nonsymmetric interior penalty discontinuous Galerkin methods, including several model problems with complicated, highcontrast jumps in the coefficients. The solver is robust with respect to an increase in the polynomial degree of the discontinuous Galerkin approximation space (at least up to degree 6), computationally efficient, and it is affected only mildly by the coefficient jumps and by the mesh size h (i.e. O(log h −1) number of iterations).
Multigrid Preconditioner for Nonconforming Discretization of Elliptic Problems with Jump Coefficients
, 1107
"... Abstract In this paper, we present a multigrid preconditioner for solving the linear system arising from the piecewise linear nonconforming CrouzeixRaviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses the standard conforming subspaces as coarse sp ..."
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Abstract In this paper, we present a multigrid preconditioner for solving the linear system arising from the piecewise linear nonconforming CrouzeixRaviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses the standard conforming subspaces as coarse spaces. Numerical tests show both robustness with respect to the jump in the coefficient and nearoptimality with respect to the number of degrees of freedom. 1
Domain Decomposition Preconditioners for HigherOrder Discontinuous Galerkin Discretizations
, 2011
"... Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, hi ..."
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Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higherfidelity simulations remains. Present day CFD simulations are limited by the lack of an efficient highfidelity solver able to take advantage of the massively parallel architectures of modern day supercomputers. A higherorder hybridizable
A ROBUST MULTIGRID METHOD FOR DISCONTINUOUS GALERKIN DISCRETIZATIONS OF STOKES AND LINEAR ELASTICITY EQUATIONS
"... Abstract. We consider multigrid methods for discontinuous Galerkin (DG) H(div, Ω)conforming discretizations of the Stokes equation. We first describe a simple Uzawa iteration for the solution of the Stokes problem, which requires a solution of a nearly incompressible linear elasticity problem on ev ..."
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Abstract. We consider multigrid methods for discontinuous Galerkin (DG) H(div, Ω)conforming discretizations of the Stokes equation. We first describe a simple Uzawa iteration for the solution of the Stokes problem, which requires a solution of a nearly incompressible linear elasticity problem on every iteration. Then, based on special subspace decompositions of H(div, Ω), as introduced in [J. Schöberl. Multigrid methods for a parameter dependent problem in primal variables. Numerische Mathematik, 84(1):97–119, 1999], we analyze variable Vcycle and Wcycle multigrid methods with nonnested bilinear forms. We prove that these algorithms are robust, and their convergence rates are independent of the material parameters such as Poisson ratio and of the mesh size. 1.
FAST SOLVERS FOR THE SYMMETRIC IPDG DISCRETIZATION OF SECOND ORDER ELLIPTIC PROBLEMS
"... Abstract. In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a s ..."
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Abstract. In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a stable decomposition of the piecewise polynomial discontinuous finite element space of arbitrary order into a linear conforming space and a space containing high frequency components. To derive such decomposition, we introduce a novel interpolation operator which projects piecewise polynomials of arbitrary order to continuous piecewise linear functions. We prove that this operator is stable which allows us to derive the required space decomposition easily. Moreover, we prove that both the condition number of the preconditioned system and the convergent rate of the iterative method are independent of the mesh size. Numerical experiments are also shown to confirm these theoretical results. Key words. Discontinuous Galerkin methods, iterative method, preconditioner. 1.
Powered by TCPDF (www.tcpdf.org) UNIFORMLY STABLE DISCONTINUOUS GALERKIN DISCRETIZATION AND ROBUST ITERATIVE SOLUTION METHODS FOR THE BRINKMAN PROBLEM
"... Uniformly stable discontinuous Galerkin discretization and robust iterative ..."
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Uniformly stable discontinuous Galerkin discretization and robust iterative
Received...
"... The incompressible Stokes equations are a widelyused model of viscous or tightly confined flow in which convection effects are negligible. In order to strongly enforce the conservation of mass at the element scale, special discretization techniques must be employed. In this paper, we consider a dis ..."
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The incompressible Stokes equations are a widelyused model of viscous or tightly confined flow in which convection effects are negligible. In order to strongly enforce the conservation of mass at the element scale, special discretization techniques must be employed. In this paper, we consider a discontinuous Galerkin (DG) approximation in which the velocity field is H(div,Ω)conforming and divergencefree, based on the BDM1 finiteelement space, with complementary space (P0) for the pressure. Due to the saddlepoint structure and the nature of the resulting variational formulation, the linear systems can be difficult to solve. Therefore, specialized preconditioning strategies are required in order to efficiently solve these systems. We compare the effectiveness of two families of preconditioners for saddlepoint systems when applied to the resulting matrix problem. Specifically, we consider blockfactorization techniques, in which the velocity block is preconditioned using geometric multigrid, as well as fullycoupled monolithic multigrid methods. We present parameter study data and a serial timing comparison, and we show that a monolithic multigrid preconditioner using BraessSarazin style relaxation provides the fastest time to solution for the test problem considered. Copyright c©
MULTILEVEL PRECONDITIONERS FOR REACTIONDIFFUSION PROBLEMS WITH DISCONTINUOUS COEFFICIENTS
"... ABSTRACT. In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reactiondiffusion equations. Specifically, we consider the multilevel preconditioners for solving the linear systems arising from the ..."
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ABSTRACT. In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reactiondiffusion equations. Specifically, we consider the multilevel preconditioners for solving the linear systems arising from the linear finite element approximation of the problem, where both diffusion and reaction coefficients are piecewiseconstant functions. We discuss in detail the influence of both the discontinuous reaction and diffusion coefficients to the performance of the classical BPX and multigrid Vcycle preconditioner. 1.