Abstract
Let Ω be an open region in ℝd, d ≥ 2, that is diffeomorphic to 𝔹d. Consider solving −Δu + γu = 0 on Ω with the Neumann boundary condition \(\frac {\partial u}{\partial \mathbf {n}}=b\left (\cdot ,u\right ) \) over ∂Ω. The function b is a nonlinear function of u. The problem is reformulated in a weak form, and then a spectral Galerkin method is used to create a sequence of finite dimensional nonlinear problems. An error analysis shows that under suitable assumptions, the solutions of the finite dimensional problems converge to those of the original problem. To carry out the error analysis, the original problem and the spectral method is converted to a nonlinear integral equation over H1/2 (Ω) , and the reformulation is analyzed using tools for solving nonlinear integral equations. Numerical examples are given to illustrate the method. In our error analysis, we assume the existence and local uniqueness of a solution. For the case of three dimensions and a nonlinearity b that is given by the Stefan–Boltzmann law, we will provide an existence proof in the final section.
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References
Amann, H.: Nonlinear elliptic equations with nonlinear boundary conditions. In: Eckhaus, W. (ed.) New Developments in Differential Equations, pp. 43–63. North-Holland (1976)
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. Numerical Algorithms 67, 691–711 (2014)
Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework. 3rd edn. Springer, Berlin (2009)
Atkinson, K., Han, W.: An Introduction to Spherical Harmonics and Approximations on the Unit Sphere. Springer, Berlin (2012)
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., Chien, D., Hansen, O.: A spectral method for nonlinear elliptic equations. Numerical Algorithms 74, 797–819 (2017)
Aubin, J.-P.: Approximation of Elliptic Boundary-Value Problems. Wiley-Interscience, Hoboken (1972)
Aubin, J.-P.: Applied Functional Analysis. 2nd edn. Wiley-Interscience, Hoboken (2000)
Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland (1978)
Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, Berlin (2013)
Delfour, M., Payre, G., Zolésios, J.-P.: Approximation of nonlinear problems associated with radiating bodies in space. SIAM J. Numerical Anal. 24, 1077–1094 (1987)
Dunkl, C., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge (2001)
Garabedian, P.: Partial Differential Equations. Wiley, Hoboken (1964)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Pub (1985)
Li, H., Xu, Y.: Spectral approximation on the the unit ball. SIAM J. Num. Anal. 52, 2647–2675 (2014)
Liou, K.-N.: An Introduction to Atmospheric Radiation. Academic Press, Cambridge (1980)
Krasnoseľskii, M.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon press, Oxford (1964)
Krasnoseľskii, M.: Positive Solutions of Operator Equations. Noordhoff (1964)
Marcus, M., Mizel, V.: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rat. Mech. Anal. 45, 294–320 (1972)
Maz’ya, V.: Sobolev Spaces. Springer, Berlin (1985)
Mikhlin, S.: Mathematical Physics, an Advanced Course. North-Holland (1970)
Osborn, J.: Spectral approximation for compact operators. Math. Comput. 29, 712–725 (1975)
Roach, G.: Green’s Functions. 2nd edn. Cambridge University Press, Cambridge (1982)
Stroud, A.: Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Eaglewood Cliffs (1971)
Treves, F.: Topological Vector Spaces, Distributions, and Kernels. Dover Publications, Mineola (2006)
Xu, Y.: Lecture notes on orthogonal polynomials of several variables. In: Advances in the Theory of Special Functions and Orthogonal Polynomials, pp. 135–188. Nova Science Publishers (2004)
Zeidler, E.: Nonlinear Functional Analysis and its Applications: I. Springer, Berlin (1986)
Zeidler, E.: Nonlinear Functional Analysis and its Applications: II/B. Springer, Berlin (1990)
Zeidler, E.: Nonlinear Functional Analysis and its Applications III. Springer, Berlin (1985)
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For helpful discussions, we thank Weimin Han and Gerhard Strohmer.
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Appendix
Appendix
Defining surface normals and Jacobian for a general surface. This is well-known in the literature, but we include it for convenience. For notational simplicity in the mapping \({\Phi }:\overline {\mathbb {B}}^{d}\underset {onto}{\overset {1-1}{\longrightarrow }}\overline {{\Omega }}\), let x be replaced by (x, y, z), and s be replaced by (s, t, u). Write
For derivatives, use the shorthand notation
with similar notation for t and u.
For the surface Jacobian |Jbdy (x, y, z)| used in the change of variables expression (2.11),
The normal at (s, t, u) = Φ (x, y, z), call it N (s, t, u), is given by
As an example, consider the ellipsoidal mapping
Then
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Atkinson, K., Chien, D. & Hansen, O. A spectral method for an elliptic equation with a nonlinear Neumann boundary condition. Numer Algor 81, 313–344 (2019). https://doi.org/10.1007/s11075-018-0550-y
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DOI: https://doi.org/10.1007/s11075-018-0550-y