ASYMPTOTIC EXPANSIONS OF THE EIGENVALUES OF BOUNDARY VALUE PROBLEMS FOR THE LAPLACE OPERATOR IN DOMAINS WITH SMALL HOLES

<p>When treating geodetic boundary value problems in gravity field studies, the geometry of the physical surface of the Earth may be seen in relation to the structure of the Laplace operator. Similarly as in other branches of engineering and mathematical physics a transformation of coordinates is used that offers a possibility to solve an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. The Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth substantially differs from a sphere or an oblate ellipsoid of revolution, even if these are optimally fitted. The situation may be more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces. The structure of the Laplace operator, however, is more complicated in this case and in a sense it represents the topography of the physical surface of the Earth. The Green&#8217;s function method together with the method of successive approximations is used for the solution of geodetic boundary value problems expressed in terms of new coordinates. The structure of iteration steps is analyzed and if useful, it is modified by means of the integration by parts. Subsequently, the individual iteration steps are discussed and interpreted.</p>

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Steklov approximations of Green’s functions for Laplace equations

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-09-2019-0357 ◽

2020 ◽

Vol 39 (4) ◽

pp. 991-1003 ◽

Cited By ~ 1

Author(s):

Manki Cho

Keyword(s):

Boundary Value Problems ◽

Laplace Operator ◽

Green's Functions ◽

Green’S Functions ◽

Correction Term ◽

Boundary Value ◽

Content Type ◽

The Laplace Operator ◽

Steklov Eigenfunctions ◽

Boundary Correction

Purpose This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator. Design/methodology/approach The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary. Findings The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary. Originality/value This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

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Very weak solutions of boundary value problems for the Laplace operator and the Lamé system on polyhedral domains in ℝ3

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500028584 ◽

1994 ◽

Vol 124 (4) ◽

pp. 645-672

Author(s):

Ding Hua

Keyword(s):

Boundary Value Problems ◽

Laplace Operator ◽

Weak Solutions ◽

Boundary Value ◽

Very Weak Solutions ◽

Nonsmooth Data ◽

Lamé System ◽

Variational Solutions ◽

Lame System ◽

The Laplace Operator

The notion of very weak solutions is introduced in this paper in order to solve the boundary value problems for the Laplace operator and for the Lamé system with nonsmooth data in polyhedral domains. A continuity theorem is given for variational solutions of the above problems. This result may be used to solve problems with concentrated loads.

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