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Keywords:
fourth-order; biharmonic operator; Laplace operators; Jacobi semi- iterative; Richardson; A.D.I.; fast Fourier transform; SIMD machine
fourth-order; biharmonic operator; Laplace operators; Jacobi semi- iterative; Richardson; A.D.I.; fast Fourier transform; SIMD machine
Summary:
The numerical solution of the model fourth-order elliptic boundary value problem in two dimensions is presented. The iterative procedure in which the biharmonic operator is splitted into two Laplace operators is used. After formulating the finite-difference approximation of the procedure, a formula for the evaluation of the transformed iteration vectors is developed. The Jacobi semi-iterative, Richardson and A.D.I. iterative Poisson solvers are applied to compute one transformed iteration vector. By the efficient use of the decomposition property of the corresponding iteration matrices, the fast Fourier transform algorithm needs to be applied twice in the evaluation of one iteration vector. The asymptotic number of operations for the sequential computation is $5n^2 log_2 n$, where $n^2$ is the number of interior grid points in the unit square. The result of$7 \ log_2 \ n$ parallel steps for the parallel computation on an SIMD machine with $n^2$ processors is so far the best one.
References:
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