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.
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Gwinner, J., Stephan, E.P. (2018). A Fourier Series Approach. In: Advanced Boundary Element Methods. Springer Series in Computational Mathematics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-319-92001-6_3
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DOI: https://doi.org/10.1007/978-3-319-92001-6_3
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