Abstract
Let Ω be an open, simply connected, and bounded region in ℝd, 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.
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Communicated by Yuesheng Xu.
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Atkinson, K., Chien, D. & Hansen, O. A spectral method for elliptic equations: the Dirichlet problem. Adv Comput Math 33, 169–189 (2010). https://doi.org/10.1007/s10444-009-9125-8
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DOI: https://doi.org/10.1007/s10444-009-9125-8