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The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation ☆

Saridakis Ioannis, Sifalakis Anastasios, Fokas, A. S., 1952-, Fulton, Ruth, 1887-1948

Απλή Εγγραφή


URIhttp://purl.tuc.gr/dl/dias/883CE8FF-45B1-42C0-939F-9B81A7C70C88-
Αναγνωριστικόhttps://doi.org/10.1016/j.cam.2007.07.012-
Γλώσσαen-
Μέγεθος25 pagesen
ΤίτλοςThe generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation ☆en
ΔημιουργόςSaridakis Ioannisen
ΔημιουργόςΣαριδακης Ιωαννηςel
ΔημιουργόςSifalakis Anastasiosen
ΔημιουργόςΣηφαλακης Αναστασιοςel
ΔημιουργόςFokas, A. S., 1952-en
ΔημιουργόςFulton, Ruth, 1887-1948en
ΕκδότηςElsevieren
ΠερίληψηA new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in Fokas [A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 53 (1997) 1411–1443]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. This is based on the analysis of the so-called global relation, an equation which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. A collocation-type numerical method for solving the global relation for the Laplace equation in an arbitrary bounded convex polygon was introduced in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. Here, by choosing a different set of the “collocation points” (values for k), we present a significant improvement of the results in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. The new collocation points lead to well-conditioned collocation methods. Their combination with sine basis functions leads to a collocation matrix whose diagonal blocks are point diagonal matrices yielding efficient implementation of iterative methods; numerical experimentation suggests quadratic convergence. The choice of Chebyshev basis functions leads to higher order convergence, which for regular polygons appear to be exponential.en
ΤύποςPeer-Reviewed Journal Publicationen
ΤύποςΔημοσίευση σε Περιοδικό με Κριτέςel
Άδεια Χρήσηςhttp://creativecommons.org/licenses/by/4.0/en
Ημερομηνία2015-10-16-
Ημερομηνία Δημοσίευσης2008-
Θεματική ΚατηγορίαGreek mathematicsen
Θεματική Κατηγορίαmathematics greeken
Θεματική Κατηγορίαgreek mathematicsen
Βιβλιογραφική ΑναφοράA. Sifalakis, A.S. Fokas, S.R. Fulton, Y. G. Saridakis, “The generalized dirichlet- neumann map for linear elliptic PDEs and its numerical implementation," J. Comp. & Applied Math., vol. 219,no. 1, pp. 9-34, 2008. doi:10.1016/j.cam.2007.07.012en

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