Numerical studies of the Steklov eigenvalue problem via conformal mappings

Alhejaili, Weaam; Kao, Chiu‐Yen

doi:10.1016/j.amc.2018.11.048

Cited by 8 publications

(4 citation statements)

References 26 publications

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“…The first method relies on the implementation in [2], where they used a conformal map, which maps the unit disk to the domain, with which they were able to reformulate the Steklov boundary problem. Using Fourier series they achieved very accurate eigenvalues, but they did not provide a method to evaluate the corresponding eigenfunctions.…”

Section: First Methods (Conformal Method)mentioning

confidence: 99%

“…This paper is organized as follows. In Section 2 we first define the Steklov problem in R 2 and show asymptotics for the eigenvalues, regarding the smoothness of the boundary, and the orthonormality of the eigenfunctions. Subsequently, we consider two higher order layer potentials, that is the single layer potential and Neumann-Poincaré operator, and their mapping properties in H p .…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we first consider three methods to numerically compute the eigenpairs. The first method is based on the results of [2], which provides the eigenvalues through a conformal map. We further develop it to also obtain the eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems

Imeri¹,

Nigam²

2022

Preprint

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In this paper we revisit an approach pioneered by Auchmuty [5] to approximate solutions of the Laplace-Robin boundary value problem. We demonstrate the efficacy of this approach on a large class of non-tensorial domains, in contrast with other spectral approaches for such problems.We establish a spectral approximation theorem showing an exponential fast numerical evaluation with regards to the number of Steklov eigenfunctions used, for smooth domains and smooth boundary data. A polynomial fast numerical evaluation is observed for either non-smooth domains or non-smooth boundary data. We additionally prove a new result on the regularity of the Steklov eigenfunctions, depending on the regularity of the domain boundary.We describe three numerical methods to compute Steklov eigenfunctions.

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Section: First Methods (Conformal Method)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems

Imeri¹,

Nigam²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…One objective is to optimize the shape of the boundary to achieve higher Steklov eigenvalues, and recent discoveries show that a rotationsymmetric, star-like domain with k spikes maximizes the k-th Steklov eigenvalues amongst perturbations of the disc [3,5]. We also want to mention a reformulation of the Laplace equation with Steklov boundary condition in [23] and existence results and bounds to the eigenvalues for the Laplace equation with mixed Steklov-Dirichlet boundary conditions in [1,12].…”

Section: Introductionmentioning

confidence: 99%

Optimization of Steklov-Neumann eigenvalues

Ammari

Imeri

Nigam

2020

Journal of Computational Physics

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This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem. Mathematics Subject Classification (MSC2000). 35R30, 35C20.Keywords. Steklov eigenvalue problem, boundary integral operators, mixed boundary conditions, perturbations formulas.

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Iterative algorithm for the conformal mapping function from the exterior of a roadway to the interior of a unit circle

Chang

Pang

et al. 2022

Arch Appl Mech

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Numerical studies of the Steklov eigenvalue problem via conformal mappings

Cited by 8 publications

References 26 publications

A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems

A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems

Optimization of Steklov-Neumann eigenvalues

Iterative algorithm for the conformal mapping function from the exterior of a roadway to the interior of a unit circle

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