A Spectral Method for the Biharmonic Equation

Let Ω be an open, simply connected, and bounded region in ℝ d $$\mathbb {R}^{d}$$ , d ≥ 2, with a smooth boundary ∂Ω that is homeomorphic to 𝕊 d − 1 $$\mathbb {S}^{d-1}$$ . Consider solving Δ 2 u + 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 ∈ C ∞ Ω ¯ $$u\in C^{\infty }\left ( \overline {\varOmega }\right ) $$ and assuming ∂Ω is a C ∞ boundary, the convergence of u − u n H 2 Ω $$\left \Vert u-u_{n}\right \Vert _{H^{2}\left ( \varOmega \right ) }$$ to zero is faster than any power of 1∕n. Numerical examples illustrate experimentally an exponential rate of convergence.