this paper we will derive a 3-G type theorem as in (1) but with G 1;n replaced by the Green function G m;n for the m-polyharmonic operator with Dirichlet boundary conditions and with

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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.

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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 right-hand 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 Ciarlet-Raviart mixed finite element method.

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...

Efficient parallel solvers for the biharmonic equation Milan D. Mihajlovi'c1 and David J. Silvester2 Abstract. We examine the convergence characteristics and performance of parallelised Krylov subspace solvers applied to the linear algebraic systems that arise from low-order mixed finite element approximation of the biharmonic problem. Our strategy results in preconditioned systems that have nearly optimal eigenvalue distribution, which consists of a tightly clustered set together with a small number of outliers. We implement the preconditioner operator in a "blackbox " fashion using publicly available parallelised sparse direct solvers and multigrid solvers for the discrete Dirichlet Laplacian. We present convergence and timing results that demonstrate efficiency and scalability of our strategy when implemented on contemporary computer architectures.

Introduction Like Boggio [Bo1] and Hadamard [Ha] (1901/08) one might conjecture that positive data f # 0, # # 0, # # 0 in the clamped plate equation 8 > < > : (-#) 2 u = f in# , u|## = , # ## u|## = #, (1) yield positive solutions u # 0. Here# # R n is the "shape of the plate" (physically relevant for n = 2), # is the exterior unit normal at # f is the (perpendicular) load, # and # are the boundary data and u is the deflection of the "plate". Most authors concentrated on the Green function G 2,n,#<F32.