A spectral method for elliptic equations: the Dirichlet problem

Let Ω be an open, simply connected, and bounded region in ? ^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 ?² and ?³ show experimentally an exponential rate of convergence.