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A Numerical Existence Proof of Nodal Lines for the First Eigenfunction of the Plate Equation
, 1996
"... . We explain a numerical procedure to compute error bounds in H 2 0 (\Omega ) for eigenfunctions of elliptic eigenvalue problems of fourth order. Therefore, we compute a finiteelement approximation and an upper bound for the defect in H \Gamma2 (\Omega ). Then, a theorem of Kato, eigenvalue inc ..."
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Cited by 11 (3 self)
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. We explain a numerical procedure to compute error bounds in H 2 0 (\Omega ) for eigenfunctions of elliptic eigenvalue problems of fourth order. Therefore, we compute a finiteelement approximation and an upper bound for the defect in H \Gamma2 (\Omega ). Then, a theorem of Kato, eigenvalue inclusions and explicit embedding constants yield a pointwise error bound for the approximation. In order to control rounding errors, we use interval arithmetic. As an application, we prove the existence of nodal lines for the first eigenfunction of the clamped plate and for the buckling plate in a square. AMS Symbol classification: 65N25 Key words: biharmonic operator, eigenfunction enclosures, plate equation, nodal lines, embedding constants The numerical computations in BauerReiss [2] and HackbuschHoffmann [12] strongly indicate the existence of nodal lines for the first eigenfunction of the clamped plate \Delta 2 u \Gamma u = 0; u 2 H 2 0 (\Omega ); \Omega = (0; 1) 2 : Here, ...
Asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle, preprint
, 1996
"... We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper b ..."
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Cited by 7 (0 self)
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We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small. 1.
Multigrid Solution Of Automatically Generated High Order Discretizations For The Biharmonic Equation
, 1998
"... . In this work, we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil. The solution and its first derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporated na ..."
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Cited by 4 (0 self)
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. In this work, we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil. The solution and its first derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporated naturally. Since the approximations use the 9 point compact stencil, no special formulas are needed near the boundaries. Both second order and fourth order discretizations are derived. The fourth order approximations produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled by the boundary condition. The method suffers from slow convergence when classical iteration methods such as GaussSeidel or SOR are employed. In order to alleviate this problem we propose several multigrid techniques which exhibit grid independent convergence and solve the biharmonic equation in a small amount of computer time. Test results from thr...
A Note on High Precision Solutions of Two Fourth Order Eigenvalue Problems
, 1998
"... We solve the biharmonic eigenvalue problem \Delta 2 u = u and the buckling plate problem \Delta 2 u = \Gamma\Deltau on the unit square using a highly accurate spectral LegendreGalerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign cha ..."
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Cited by 3 (0 self)
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We solve the biharmonic eigenvalue problem \Delta 2 u = u and the buckling plate problem \Delta 2 u = \Gamma\Deltau on the unit square using a highly accurate spectral LegendreGalerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign changes are computed and the results show that the eigenfunction exhibits a self similar pattern as one approaches the corner. The amplitudes of the extremal values and the coordinates of their location as measured from the corner are reduced by constant factors. These results are compared with the known asymptotic expansion of the solution near a corner. This comparison shows that the asymptotic expansion is highly accurate already from the first sign change as we have complete agreement between the numerical and the analytical results. Thus, we have an accurate description of the eigenfunction in the entire domain. AMS Subject Classification: 65N25. Key words: Biharmonic operator, eigenvalu...
An efficient direct solver for a class of mixed finite element problems
 SCHOOL OF COMPUTER STUDIES RESEARCH REPORT 99.03 (UNIVERSITY OF LEEDS
, 1999
"... In this paper we present an efficient, accurate and parallelizable direct method for the solution of the (indefinite) linear algebraic systems that arise in the solution of fourth order partial differential equations (PDEs) using mixed finite element approximations. The method is intended particular ..."
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Cited by 3 (1 self)
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In this paper we present an efficient, accurate and parallelizable direct method for the solution of the (indefinite) linear algebraic systems that arise in the solution of fourth order partial differential equations (PDEs) using mixed finite element approximations. The method is intended particularly for use when multiple righthand sides occur, and when high accuracy is required in these solutions. The algorithm is described in some detail and its performance is illustrated through the numerical solution of a biharmonic eigenvalue problem where the smallest eigenpair is approximated using inverse iteration after discretization via the CiarletRaviart mixed finite element method.
On the Accurate Finite Element Solution of a Class of Fourth Order Eigenvalue Problems
, 1995
"... 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 ..."
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Cited by 2 (0 self)
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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 LegendreGalerkin 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...