# A VLSI multigrid poisson solver amenable to biharmonic equation

## Abstract

A VLSI parallel algorithm for solving discretized Poisson equation on an *NxN* grid is proposed. A standard multigrid algorithm is adopted which allows a parallel solution of this problem in *T=O(logN)* time steps. A special network consisting of *N × N* processor elements and of *O(NlogN)* interconnection lines in each direction results in a design the area of which is *O(N*^{2}log^{2}N). Thus, the *AT*^{2} estimation for this Poisson solver is *O(N*^{2}log^{4}N) which improves the best result known until now by factor of *O(N/logN)*.

An application of the multigrid Poisson solver is made to a VLSI semidirect biharmonic solver. The VLSI layout needs an area *A=O(N*^{3}logN) and the time of algorithm is *O(√Nlog*^{2}N). The total complexity for the VLSI biharmonic solver is *AT*^{2}=O(N^{4}log^{5}N) which is of the same order as for the best algorithms developed until now.

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