A spectral method for elliptic equations: the Dirichlet problem.

Atkinson, Kendall; Chien, David; Hansen, Olaf

Let Ω be an open, simply connected, and bounded region in ℝ d, d ≥ 2, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation Lu = f over Ω with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B; and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials u n of degree ≤ n that is convergent to u. The transformation from Ω to B requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}( \overline{\Omega})$ and assuming $\partial\Omega$ is a C ∞ boundary, the convergence of $\left\Vert u-u_{n}\right\Vert _{H^{1}}$ to zero is faster than any power of 1/ n. Numerical examples in ℝ2 and ℝ3 show experimentally an exponential rate of convergence.

SPECTRAL geometry; DIRICHLET problem; ELLIPTIC differential equations; GALERKIN methods; ORTHOGRAPHIC projection; BOUNDARY value problems

Advances in Computational Mathematics, 2010, Vol 33, Issue 2, p169

1019-7168

Academic Journal

10.1007/s10444-009-9125-8