## On the Accurate Finite Element Solution of a Class of Fourth Order Eigenvalue Problems (1995)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Brown95onthe,

author = {B.M. Brown and E. B. Davies and P.K. Jimack and M.D. Mihajlovic},

title = {On the Accurate Finite Element Solution of a Class of Fourth Order Eigenvalue Problems},

year = {1995}

}

### OpenURL

### Abstract

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corner on domain boundaries. Recent computational results of Bjrstad and Tjstheim [4], using a highly accurate spectral Legendre-Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained...

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Citation Context ...1 0 (Ω) = ψ ∈ H 1 (Ω) : ψ|∂Ω = ∂ψ ∣ ∂n Consider a family of regular and quasi-uniform triangulations, T h (where h is the diameter of the largest triangle), of the domain Ω (see, for example, Ciarlet =-=[10]-=-), then we may define the space ∣ ∂Ω = 0 S h m = {p ∈ C0 ( ¯ Ω) : p|T ∈ Pm(T), ∀T ∈ T h }, where Pm(T) is the space of polynomials of degree at most m over triangle T. From this we may define (Ω). The... |

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Citation Context ...cus, without affecting either the dimension or the sparsity of the discrete linear system (10). The results presented in Section 3 are in agreement with the asymptotic theory that appears in [6],[12],=-=[17]-=-,[18] and also provide quantitative estimates of the positions of a number of sign changes and extremal points for various values of the internal angle θ. In Section 4 we extend the numerical investig... |

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Citation Context ...imulations such as these, including mesh convergence and the use of rigorous enclosure techniques. 1.1 Oscillatory properties of eigenfunctions For each of the two problems (1) and (2) it is shown in =-=[6]-=-,[12] that in the neighbourhood of a corner of ∂Ω with sufficiently small internal angle θ any eigenfunction changes sign an infinite number of times. Moreover the ratio of the distance from the corne... |

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Citation Context ... upon the use of conforming C0 Lagrange finite elements (see [21],[25] for example) on some triangulation of the domain Ω. The particular method that we use is based on that of Ciarlet and Raviart in =-=[11]-=-. Let H1 (Ω) represent the usual Sobolev space of L2 (Ω) functions whose first partial derivatives are in L2 (Ω), and { ∣ } H 1 0 (Ω) = ψ ∈ H 1 (Ω) : ψ|∂Ω = ∂ψ ∣ ∂n Consider a family of regular and qu... |

19 |
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Citation Context ...zation scheme must be able to accommodate quite general domain geometries. For this reason we choose a numerical scheme which is based upon the use of conforming C0 Lagrange finite elements (see [21],=-=[25]-=- for example) on some triangulation of the domain Ω. The particular method that we use is based on that of Ciarlet and Raviart in [11]. Let H1 (Ω) represent the usual Sobolev space of L2 (Ω) functions... |

17 |
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Citation Context ...to be present when the internal angle exceeds some critical value θc. These results are special cases of more general theorems concerning higher order elliptic operators in greater than one dimension =-=[18]-=-. Their derivation is based upon an asymptotic expansion of the eigenfunction, u, centred at, and in a neighbourhood of, the corner. The leading term of such an expansion as one approaches the corner ... |

15 | Lp spectral theory of higher-order elliptic differential operators
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Citation Context ...perators. However, in contrast to the well-understood theory for second order operators, the higher order theory can be quite different and is certainly less well developed. The recent survey article =-=[14]-=- contains an account of many of the key results. The purpose of this paper is to explore, using reliable and accurate numerical techniques, some of the properties of the eigenfunctions of the biharmon... |

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Citation Context ...ng the bisector tends to 36267.550 as n → ∞. In recent years many authors have tried to verify numerically these theoretical results using a variety of computational techniques (see, for example, [2],=-=[9]-=-,[16],[24]). Since the oscillatory features occur very close to the corners and are damped out very quickly, most of these attempts have, due to discretization errors and numerical inaccuracy, failed ... |

8 |
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Citation Context ...he bisector tends to 36267.550 as n → ∞. In recent years many authors have tried to verify numerically these theoretical results using a variety of computational techniques (see, for example, [2],[9],=-=[16]-=-,[24]). Since the oscillatory features occur very close to the corners and are damped out very quickly, most of these attempts have, due to discretization errors and numerical inaccuracy, failed to fi... |

8 |
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Citation Context ...ations such as these, including mesh convergence and the use of rigorous enclosure techniques. 1.1 Oscillatory properties of eigenfunctions For each of the two problems (1) and (2) it is shown in [6],=-=[12]-=- that in the neighbourhood of a corner of ∂Ω with sufficiently small internal angle θ any eigenfunction changes sign an infinite number of times. Moreover the ratio of the distance from the corner, al... |

6 | Efficient algorithms for solving a fourth order equation with the spectral Galerkin method - Bjørstad, Tjøstheim - 1997 |

5 | On the structure of solutions \Delta u = u which satisfy the clamped plate conditions on a right angle - Coffman - 1969 |

4 | A note on high precision solutions of two fourth order eigenvalue problems
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Citation Context ...on is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corner on domain boundaries. Recent computational results of Bjørstad and Tjøstheim =-=[4]-=-, using a highly accurate spectral Legendre-Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with ... |

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- BROWN, JIMACK, et al.
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Citation Context ...ion. We solve the systems (12) using a direct sparse method based upon an initial block reduction followed by a sparse Cholesky factorization of symmetric positive definite sub-blocks as described in =-=[8]-=-. This permits re-use of the factorizations for the second and subsequent solves, which ensures that the cost of these is significantly less than that of the initial solution of (12). An additional ad... |

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2 | A numerical investigation of some problems associated with the biharmonic operator - Mihajlovi'c - 1999 |