Mixed boundary value problems in plane domains with corners

Wigley, Neil M.

doi:10.1007/bf01109747

Cited by 49 publications

(21 citation statements)

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“…Just as in the case of a smooth boundary, in addition to methods related to H61der classes, as employed in the present paper, there are other approaches; we mention in particular Sobolev space methods as considered in the reviews by Grisvard [20], [21], then Kondratiev's extension [26] of the SobolevSlobodeckii space method by Eskin [13] and Vigik [48] (cf. also [3]), furthermore a function theoretic approach by Lewy [29] and his school (see, for instance, Wigley [53]) and, finally, a recent method by Simon [42] based on geometric measure theory.…”

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confidence: 99%

On Solutions of Elliptic Equations Satisfying Mixed Boundary Conditions

Azzam¹,

Kreyszig²

1982

SIAM J. Math. Anal.

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confidence: 99%

On Solutions of Elliptic Equations Satisfying Mixed Boundary Conditions

Azzam¹,

Kreyszig²

1982

SIAM J. Math. Anal.

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“…It is known [45,46] that, under our assumption that the curve is piecewise smooth, for any given integer N and any given positive number ε the eigenfunctions in equation (1) can be expressed in the form…”

Section: Singularities In Eigenfunctions and Integral Equation Densitiesmentioning

confidence: 99%

A boundary integral algorithm for the Laplace Dirichlet–Neumann mixed eigenvalue problem

Akhmetgaliyev

Bruno

Nigam

2015

Journal of Computational Physics

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We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course, (slight modifications of) our algorithms are also applicable to the pure Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of an ansatz based on the single layer boundary operator, the Zaremba eigenproblem is transformed into a nonlinear equation for the eigenvalue μ. For smooth domains the singular structure at Dirichlet-Neumann junctions is incorporated as part of our corresponding numerical algorithm-which otherwise relies on use of the cosine change of variables, trigonometric polynomials and, to avoid the Gibbs phenomenon that would arise from the solution singularities, the Fourier Continuation method (FC). The resulting numerical algorithm converges with high order accuracy without recourse to use of meshes finer than those resulting from the cosine transformation. For non-smooth (Lipschitz) domains, in turn, an alternative algorithm is presented which achieves highorder accuracy on the basis of graded meshes. In either case, smooth or Lipschitz boundary, eigenvalues are evaluated by searching for zero minimal singular values of a suitably stabilized discrete version of the single layer operator mentioned above. (The stabilization technique is used to enable robust non-local zero searches.) The resulting methods, which are fast and highly accurate for high-and low-frequencies alike, can solve extremely challenging two-dimensional Dirichlet, Neumann and Zaremba eigenproblems with high accuracies in short computing times-enabling, in particular, evaluation of thousands of eigenvalues and corresponding eigenfunctions for a given smooth or non-smooth geometry with nearly full double-precision accuracy.

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“…Da zur Untersuchung von Funktionen in Gebieten mit Ecken gewichtete Funktionenräume sehr gut geeignet sind, vergleiche beispielsweise [13], [14], [29], benutzen wir die folgenden Funktionenklassen, die auch in [20] und [21] verwendet wurden. Definition 2.1.…”

Section: Zurückführung Des Neumannproblems Auf Ein Dirichletproblemunclassified

Zum Neumannschen Randwertproblem quasilinearer elliptischer Differentialgleichungen zweiter Ordnung in Gebieten mit Ecken

Knobloch

2000

Math. Nachr.

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Mixed boundary value problems in plane domains with corners

Cited by 49 publications

References 6 publications

On Solutions of Elliptic Equations Satisfying Mixed Boundary Conditions

On Solutions of Elliptic Equations Satisfying Mixed Boundary Conditions

A boundary integral algorithm for the Laplace Dirichlet–Neumann mixed eigenvalue problem

Zum Neumannschen Randwertproblem quasilinearer elliptischer Differentialgleichungen zweiter Ordnung in Gebieten mit Ecken

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