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The Dirichlet problem on quadratic surfaces

Authors: Sheldon Axler, Pamela Gorkin and Karl Voss
Journal: Math. Comp. 73 (2004), 637-651
MSC (2000): Primary 31B05, 31B20
Published electronically: June 10, 2003
MathSciNet review: 2031398
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Abstract: We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in $\mathbf {R}^n$ such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in $\mathbf {R}^n$ can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.

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Additional Information

Sheldon Axler
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132
MR Author ID: 201457
ORCID: 0000-0003-1733-6080

Pamela Gorkin
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
MR Author ID: 75530

Karl Voss
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Keywords: Laplacian, Dirichlet problem, harmonic, ellipsoid, polynomial, quadratic surface
Received by editor(s): November 11, 2002
Published electronically: June 10, 2003
Additional Notes: The first author was supported in part by the National Science Foundation
Article copyright: © Copyright 2003 American Mathematical Society