A spectral method for the biharmonic equation

Abstract

Let Ω be an open, simply connected, and bounded region in ℝd, d ≥ 2, with a smooth boundary ∂Ω that is homeomorphic to Sd-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 ε C∞ (Ω) and assuming ∂Ω is a C∞ boundary, the convergence of ||u-un||H2(Ω) to zero is faster than any power of 1/n. Numerical examples illustrate experimentally an exponential rate of convergence.