A spectral method for elliptic equations: the Neumann problem

Authors:
Kendall Atkinson;Olaf Hansen;David Chien
Affiliations:
Departments of Mathematics & Computer Science, The University of Iowa, Iowa City, USA;Department of Mathematics, California State University San Marcos, San Marcos, USA;Department of Mathematics, California State University San Marcos, San Marcos, USA
Venue:
Advances in Computational Mathematics
Year:
2011

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Abstract

Let 驴 be an open, simply connected, and bounded region in 驴 d , d驴驴驴2, and assume its boundary 驴驴 is smooth. Consider solving the elliptic partial differential equation 驴驴Δu驴+驴驴u驴=驴f over 驴 with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable, and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.