A Partition of Unity Method with Penalty for Fourth Order Problems

Davis, Connor

doi:10.1007/s10915-013-9795-8

Cited by 15 publications

(3 citation statements)

References 29 publications

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“…An alternative is to employ generalized finite element methods [28,2]. This was carried out in [9] using a flat-top partition of unity method (PUM) from [21,30,29,14]. Below we recall some basic facts concerning the PUM in [9].…”

Section: A Partition Of Unity Methodsmentioning

confidence: 99%

Additive Schwarz preconditioners for the obstacle problem of clamped Kirchhoff plates

Brenner¹,

Davis²,

Sung³

2019

etna

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When the obstacle problem of clamped Kirchhoff plates is discretized by a partition of unity method, the resulting discrete variational inequalities can be solved by a primal-dual active set algorithm. In this paper we develop and analyze additive Schwarz preconditioners for the systems that appear in each iteration of the primal-dual active set algorithm. Numerical results that corroborate the theoretical estimates are also presented.

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Section: A Partition Of Unity Methodsmentioning

confidence: 99%

Additive Schwarz preconditioners for the obstacle problem of clamped Kirchhoff plates

Brenner¹,

Davis²,

Sung³

2019

etna

Self Cite

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“…However, spurious solutions may occur in some situations. The conforming finite element methods including Argyris elements [2] and the partition of unity finite elements [11], require globally continuously differentiable finite element spaces, which are difficult to construct and implement. The third type of approaches use non-conforming finite element methods, such as Adini elements [1], Morley elements [19,21,25] and the ordinary C 0 -interior penalty Galerkin method [26].…”

Section: Introductionmentioning

confidence: 99%

Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

Zhang

2019

Numer Algor

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In this paper we propose and analyze spectral-Galerkin methods for the Stokes eigenvalue problem based on the stream function formulation in polar geometries. We first analyze the stream function formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk to a sequence of equivalent onedimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one dimensional problem can be obtained. Further, we extend our method to the non-separable Stokes eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.

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“…However, most of the existing studies were concerned with the second-order elliptic eigenvalue problems and there are relatively few works treating the biharmonic eigenvalue problems. In recent years, the numerical methods of eigenvalue problems were mainly based on finite element methods which include conforming finite elements [3,11,20,25], nonconforming finite elements [1,18,21,23], and mixed finite elements [5,10,17,19]. For the conforming finite element method, it requires globally continuously differentiable finite element spaces, which are difficult to construct and implement (in particular for three dimensional problems).…”

Section: Introductionmentioning

confidence: 99%

A high accuracy spectral method based on min/max principle for biharmonic eigenvalue problems on a spherical domain

Luo

2016

Journal of Mathematical Analysis and Applications

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In this study, we develop a high precision spectral method based on the min/max principle for biharmonic eigenvalue problems in the spherical domain. By analyzing the orthogonal spherical harmonic and approximation and using the min/max principle, we first deduce the error estimates of approximate eigenvalues. Then we construct an appropriate set of orthogonal spherical functions contained in H 2 0 (Ω) and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate that the theoretical results are correct.

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A Partition of Unity Method with Penalty for Fourth Order Problems

Cited by 15 publications

References 29 publications

Additive Schwarz preconditioners for the obstacle problem of clamped Kirchhoff plates

Additive Schwarz preconditioners for the obstacle problem of clamped Kirchhoff plates

Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

A high accuracy spectral method based on min/max principle for biharmonic eigenvalue problems on a spherical domain

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