Abstract We report on an experimental study of the effectiveness of 13 methods to solve the systems of linear equations obtained using the Galerkin method with bicubic Hermite polynomial basis functions applied to two-dimensional elliptic partial differential equations. The study concludes that, within 99% confidence levels, the iteration methods considered provide an advantage over the usual Gauss elimination methods. The cross-over point for iteration methods becoming most efficient is usually for about an 11 by 11 grid (observed range: 7 by 7 to 17 by 17). These results support the conjecture that iteration with optimal parameter is as effective for finite element method systems of equations as it is known to be for finite difference method equations. These results are not in agreement with theoretical expectations about the asymptotic behavior of sparse matrix methods: some possible sources of the discrepancy are listed.
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