C 0 Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems

Brenner, Susanne C.; Monk, Peter; Sun, Jiguang

doi:10.1007/978-3-319-19800-2_1

Cited by 44 publications

(47 citation statements)

References 22 publications

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“…There has been a long history on developing the finite element methods of the biharmonic eigenvalue problem, and many schemes have been proposed for discretization [9,11,25,36], computation of guaranteed upper and lower bounds [10,22,23,43], and adaptive method and its convergence analysis [17]. This paper is devoted to studying the multilevel efficient method of the biharmonic eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

Zhang

2017

J Sci Comput

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Abstract. In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization.Then, we construct multi-level finite element schemes by implementing the algorithm as in [33] to the nested discretizations on series of nested grids. The multi-level mixed scheme for biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.

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Section: Introductionmentioning

confidence: 99%

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

Zhang

2017

J Sci Comput

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“…The use of a penalty on the normal derivative to solve the plate eigenvalue problem is also applied in a FEM context by the C 0 -IPDG method [35]. The geometry is represented by 16 patches coupled across 16 interfaces as shown in Figure 8.…”

Section: 3mentioning

confidence: 99%

A hybrid isogeometric approach on multi-patches with applications to Kirchhoff plates and eigenvalue problems

Horger

Reali

Wohlmuth

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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We present a systematic study on higher-order penalty techniques for isogeometric mortar methods. In addition to the weak-continuity enforced by a mortar method, normal derivatives across the interface are penalized. The considered applications are fourth order problems as well as eigenvalue problems for second and fourth order equations. The hybrid coupling enables the discretization of fourth order problems in a multi-patch setting as well as a convenient implementation of natural boundary conditions. For second order eigenvalue problems, the pollution of the discrete spectrum -typically referred to as "outliers" -can be avoided.Numerical results illustrate the good behaviour of the proposed method in simple systematic studies as well as more complex multi-patch mapped geometries for linear elasticity and Kirchhoff plates.

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“…In all experiments we consider the viscosity ν = 1 and chose the penalty parameter γ = k(k + 1)/2 for k-th order RT k × Q k finite element pairs, k = 1, 2, 3. Since the eigenvalues of the Stokes problem are related to the eigenvalues of the buckling eigenvalue problem of clamped plates via the stream function formulation, we can use reference values for the eigenvalues from [6,7,31].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

Gedicke

Khan

2019

Numer. Math.

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In this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a robust residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient in a meshdependent velocity-pressure norm. We finally present some numerical examples that verify the a priori convergence rates and the reliability and efficiency of the residual based a posteriori error estimator.

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C 0 Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems

Cited by 44 publications

References 22 publications

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

A hybrid isogeometric approach on multi-patches with applications to Kirchhoff plates and eigenvalue problems

Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

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