Abstract
Let Ω be an open, simply connected, and bounded region in \(\mathbb {R}^{d}\), d ≥ 2, and assume its boundary ∂Ω is smooth and homeomorphic to \(\mathbb {S}^{d-1}\). Consider solving an elliptic partial differential equation L u = f(⋅, u) over Ω with zero Dirichlet boundary value. The function f is a nonlinear function of the solution u. The problem is converted to an equivalent elliptic problem over the open unit ball \(\mathbb {B}^{d}\) in \(\mathbb {R}^{d}\), say \(\widetilde {L}\widetilde {u} =\widetilde {f}(\cdot ,\widetilde {u})\). Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials \(\widetilde {u} _{n}\) of degree ≤ n that is convergent to \(\widetilde {u}\). The transformation from Ω to \(\mathbb {B}^{d}\) requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty } \left (\overline {\Omega }\right ) \) and assuming ∂Ω is a C ∞ boundary, the convergence of \(\left \Vert \widetilde {u} -\widetilde {u}_{n}\right \Vert _{H^{1}}\) to zero is faster than any power of 1/n. The error analysis uses a reformulation of the boundary value problem as an integral equation, and then it uses tools from nonlinear integral equations to analyze the numerical method. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to −Δu + γ u = f(u) with a zero Neumann boundary condition is also presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Atkinson, K.: The numerical evaluation of fixed points for completely continuous operators. SIAM J. Num. Anal. 10, 799–807 (1973)
Atkinson, K., Chien, D., Hansen, O.: A spectral method for elliptic equations The Dirichlet problem. Adv. Comput. Math. 33, 169–189 (2010)
Atkinson, K., Chien, D., Hansen, O.: Evaluating polynomials over the unit disk and the unit ball. Numer. Algorithm. 67, 691–711 (2014)
Atkinson, K., Han, W.: Theoretical numerical analysis: a functional analysis framework, 3rd edn. Springer-Verlag (2009)
Atkinson, K., Hansen, O.: Creating domain mappings. Electron. Trans. Numer. Anal. 39, 202–230 (2012)
Atkinson, K., Hansen, O., Chien, D.: A spectral method for elliptic equations The Neumann problem. Adv. Comput. Math. 34, 295–317 (2011)
Atkinson, K., Hansen, O., Chien, D.: A spectral method for parabolic differential equations. Numer. Algorithm. 63, 213–237 (2013)
Dunkl, C., Xu, Y.: Orthogonal polynomials of several variables. Cambridge University Press (2001)
Kot, M.: Elements of mathematical ecology. Cambridge University Press (2001)
Li, H., Xu, Y.: Spectral approximation on the unit ball. SIAM J. Num Anal. 52, 2647–2675 (2014)
Logan, B., Shepp, L.: Optimal reconstruction of a function from its projections. Duke Math. J. 42, 645–659 (1975)
Krasnoseľskii, M.: Topological methods in the theory of nonlinear integral equations. Pergamon Press (1964)
Marcus, M., Mizel, V.: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rat. Mech. Anal. 45, 294–320 (1972)
Osborn, J.: Spectral approximation for compact operators. Math. Comput. 29, 712–725 (1975)
Stroud, A.: Approximate calculation of multiple integrals. Prentice-Hall, Inc. (1971)
Zeidler, E.: Nonlinear functional analysis and its applications: II/B. Springer-Verlag (1990)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Atkinson, K., Chien, D. & Hansen, O. A spectral method for nonlinear elliptic equations. Numer Algor 74, 797–819 (2017). https://doi.org/10.1007/s11075-016-0172-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0172-1