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A Spectral Method for Elliptic Equations: The Dirichlet Problem
, 2009
"... Let be an open, simply connected, and bounded region in R d, d 2, and assume its boundary @ is smooth. Consider solving an elliptic partial di¤erential equation Lu = f over with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B; and then ..."
Abstract

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Let be an open, simply connected, and bounded region in R d, d 2, and assume its boundary @ is smooth. Consider solving an elliptic partial di¤erential equation Lu = f over 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 un of degree n that is convergent to u. The transformation from to B requires a special analytical calculation for its implementation. With su ¢ ciently smooth problem parameters, the method is shown to be rapidly convergent. For u 2 C 1 and assuming @ is a C 1 boundary, the convergence of ku unk H 1 to zero is faster than any power of 1=n. Numerical examples in R 2 and R 3 show experimentally an exponential rate of convergence. 1