A Spectral Method for the Biharmonic Equation

  • Kendall Atkinson
  • David Chien
  • Olaf Hansen


Let Ω be an open, simply connected, and bounded region in \(\mathbb {R}^{d}\), d ≥ 2, with a smooth boundary ∂Ω that is homeomorphic to \(\mathbb {S}^{d-1}\). Consider solving Δ2u + γu = f over Ω with zero Dirichlet boundary conditions. A Galerkin method based on a polynomial approximation space is proposed, yielding an approximation un. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty }\left ( \overline {\varOmega }\right ) \) and assuming ∂Ω is a C boundary, the convergence of \(\left \Vert u-u_{n}\right \Vert _{H^{2}\left ( \varOmega \right ) }\) to zero is faster than any power of 1∕n. Numerical examples illustrate experimentally an exponential rate of convergence.


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Authors and Affiliations

  1. 1.The University of IowaIowa CityUSA
  2. 2.California State University San MarcosSan MarcosUSA

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