Abstract
A two-step quadratic spline collocation method is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves fast Fourier transforms and, in comparison to its one-step counterpart, it has the advantage of requiring the solution a symmetric positive definite Schur complement system rather than a nonsymmetric one. As a consequence, the corresponding step of the new method is performed using a preconditioned conjugate gradient method. The total cost of the method on a N × N partition of the unit square is \(O(N^{2}\log N)\). To demonstrate the optimal accuracy of the method, the results of numerical experiments are provided.
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The authors are grateful for the constructive remarks and useful suggestions of an anonymous referee which helped to improve the paper.
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Bialecki, B., Fairweather, G. & Karageorghis, A. An optimal two-step quadratic spline collocation method for the Dirichlet biharmonic problem. Numer Algor 91, 1115–1143 (2022). https://doi.org/10.1007/s11075-022-01294-y
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DOI: https://doi.org/10.1007/s11075-022-01294-y
Keywords
- Biharmonic equation
- Quadratic spline collocation
- Fast Fourier transforms
- Preconditioned conjugate gradient method
- Optimal global convergence rates
- Superconvergence