12 research outputs found
Computing the function via the inverse power method
In this paper, we discuss a new iterative method for computing .
This function was introduced by Lindqvist in connection with the unidimensional
nonlinear Dirichlet eigenvalue problem for the -Laplacian. The iterative
technique was inspired by the inverse power method in finite dimensional linear
algebra and is competitive with other methods available in the literature
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift
In this paper we present an iterative method, inspired by the inverse
iteration with shift technique of finite linear algebra, designed to find the
eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet
boundary condition for arbitrary bounded domains . This
method, which has a direct functional analysis approach, does not approximate
the eigenvalues of the Laplacian as those of a finite linear operator. It is
based on the uniform convergence away from nodal surfaces and can produce a
simple and fast algorithm for computing the eigenvalues with minimal
computational requirements, instead of using the ubiquitous Rayleigh quotient
of finite linear algebra. Also, an alternative expression for the Rayleigh
quotient in the associated infinite dimensional Sobolev space which avoids the
integration of gradients is introduced and shown to be more efficient. The
method can also be used in order to produce the spectral decomposition of any
given function .Comment: In this version the numerical tests in Section 6 were considerably
improved and the Section 5 entitled "Normalization at each step" was
introduced. Moreover, minor adjustments in the Section 1 (Introduction) and
in the Section 7 (Fi nal Comments) were made. Breno Loureiro Giacchini was
added as coautho
Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions
We introduce an iterative method for computing the first eigenpair
for the -Laplacian operator with homogeneous Dirichlet
data as the limit of as , where
is the positive solution of the sublinear Lane-Emden equation
with same boundary data. The method is
shown to work for any smooth, bounded domain. Solutions to the Lane-Emden
problem are obtained through inverse iteration of a super-solution which is
derived from the solution to the torsional creep problem. Convergence of
to is in the -norm and the rate of convergence of
to is at least . Numerical evidence is
presented.Comment: Section 5 was rewritten. Jed Brown was added as autho
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