Numerical analysis of Fokas' unified method for linear elliptic PDEs

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• 作者：A.C.L. Ashton ; ¹ ; ^{a.c.l.ashton@damtp.cam.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; K.M. Crooks ^{k.m.crooks@maths.cam.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Elliptic ; Unified method ; Dirichlet eigenvalue ; Convex polygon
• 刊名：Applied Numerical Mathematics
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：104
• 期：Complete
• 页码：120-132
• 全文大小：1036 K

文摘

In this paper we examine a numerical implementation of Fokas' unified method for elliptic boundary value problems on convex polygons. Within this setting the unified method provides a reconstruction of the unknown boundary values from the known boundary data for an important class of elliptic boundary value problems. We demonstrate that this approach can be used to study the classical Dirichlet eigenvalue problem for the Laplacian on convex polygons. We show that this problem is equivalent to a nonlinear eigenvalue problem for a semi-Fredholm operator which has holomorphic dependence on the eigenvalue itself. We also study the classical Dirichlet problem for the Helmholtz operator. We provide a new Galerkin scheme for the underlying Dirichlet–Neumann map, and prove that for sufficiently regular Dirichlet data this scheme converges with spectral accuracy.