A Multilevel Preconditioner for the Interior Penalty Discontinuous Galerkin Method

Brix, Kolja; Pinto, Martin Campos; Dahmen, Wolfgang

doi:10.1137/07069691x

Cited by 25 publications

(28 citation statements)

References 20 publications

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“…Due to the symmetry of the bilinear form a T (·, ·), conjugate gradient methods may be used to solve (2.11). Multilevel preconditioners are studied in [30,19] but, for simplicity, we assume that U is computed exactly.…”

Section: The Adaptive Procedurementioning

confidence: 99%

Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method

Bonito¹,

Nochetto²

2010

SIAM J. Numer. Anal.

129

165

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Abstract. We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded, and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental to derive optimal cardinality of ADFEM. We show that ADFEM (and AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

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Section: The Adaptive Procedurementioning

confidence: 99%

Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method

Bonito¹,

Nochetto²

2010

SIAM J. Numer. Anal.

129

165

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“…For simplicity, we may take small constant penalty parameters for OPSIPG scheme. Based on the estimate (11), then we have the following lemma for (8).…”

Section: Some Estimates Of the Opsipg Schemementioning

confidence: 99%

“…The OPSIPG and WOPSIP methods produce an ill-conditioned discrete system, which results from the over-penalization terms. Fortunately, it can be remedied by a simple block-diagonal preconditioner (see [6]) and a multilevel preconditioner in [8]. Now the main question lies in how to implement the PPR technique into discontinuous Galerkin solutions under the framework of discontinuous Galerkin finite element methods.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method

Song

Zhang

2015

Journal of Computational Physics

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A polynomial preserving recovery method is introduced for over-penalized symmetric interior penalty discontinuous Galerkin solutions to a quasi-linear elliptic problem. As a post-processing method, the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns, as well as general meshes satisfying Condition ( , σ ). By means of the averaging technique, we prove the polynomial preserving recovery method for averaged solutions is superconvergent, satisfying similar estimates as those for conforming finite element methods. We deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact. Extensive numerical results consistent with our analysis are presented.

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“…The p-multigrid (synonymously spectral multigrid [1], hierarchic multigrid [2], or multi-level method [3]) algorithm originally proposed by Rönquist and Patera [1] is an iterative technique to solve hp-finite element discretizations of equations. After a dormant period of more than a decade, the method has received renewed interest in recent works by Helenbrook et al [4], Helenbrook and Atkins [5,6], Fidkowski et al [7], Oliver [8] and Luo et al [9].…”

Section: Introductionmentioning

confidence: 99%