Advanced Boundary Element Methods pp 43-62 | Cite as
A Fourier Series Approach
Abstract
The aim of this chapter is to guide the reader from elementary Fourier series expension to periodic Sobolev spaces on a simply connected smooth curve in \(\mathbb {R}^{2}\). In this tour we detail on dual spaces and compact embedding. This leads to the compactness of the double-layer operator and its adjoint. Moreover in the scale of Sobolev spaces we prove the mapping property of the single-layer and hypersingular operators. Then we treat the exterior Dirichlet problem for the Laplacian and derive its explicit solution on the unit circle in terms of the Fourier coefficients. The Fourier tour concludes with the first Gårding inequality for a bilinear form which is basic in the BEM.
References
- 171.B. Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. I. The two-dimensional case. IMA J. Numer. Anal. 20, 203–234 (2000)MathSciNetCrossRefGoogle Scholar
- 257.G.C. Hsiao, W.L. Wendland, A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58, 449–481 (1977)MathSciNetCrossRefGoogle Scholar
- 259.G.C. Hsiao, W.L. Wendland, Boundary Integral Equations. Applied Mathematical Sciences, vol. 164 (Springer, Berlin, 2008)CrossRefGoogle Scholar
- 272.J. Král, Integral Operators in Potential Theory. Lecture Notes in Mathematics, vol. 823 (Springer, Berlin, 1980)CrossRefGoogle Scholar
- 276.R. Kress, Linear Integral Equations, 3rd edn. Applied Mathematical Sciences, vol. 82 (Springer, New York, 2014)Google Scholar
- 301.V.G. Maz’ya, Boundary Integral Equations. Analysis, IV, Encyclopaedia Math. Sci., vol. 27 (Springer, Berlin, 1991), pp. 127–222Google Scholar
- 324.J.C. Nédélec, J. Planchard, Une méthode variationnelle d’ éléments finis pour la résolution numérique d’un problème extérieur dans R 3. Rev. Franc. Automat. Informat. Rech. Operat. Sér. Rouge 7, 105–129 (1973)zbMATHGoogle Scholar
- 343.S. Prössdorf, Linear Integral Equations. Analysis, IV, Encyclopaedia Math. Sci., vol. 27 (Springer, Berlin, 1991), pp. 1–125Google Scholar
- 347.J. Radon, Über die Randwertaufgabe beim logarithmischen Potential. Math.–Nat. Kl. Abt. IIa 128, 1123–1167 (1919)Google Scholar
- 356.J. Saranen, G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer Monographs in Mathematics (Springer, Berlin, 2002)Google Scholar
- 409.E.P. Stephan, W.L. Wendland, Mathematische Grundlagen der finiten Element-Methoden. Methoden und Verfahren der Mathematischen Physik [Methods and Procedures in Mathematical Physics], vol. 23 (Verlag Peter D. Lang, Frankfurt, 1982)Google Scholar