4 research outputs found
Efficient solution of symmetric eigenvalue problems from families of coupled systems
Efficient solution of the lowest eigenmodes is studied for a family of
related eigenvalue problems with common block structure. It is
assumed that the upper diagonal block varies between different versions while
the lower diagonal block and the range of the coupling blocks remains
unchanged. Such block structure naturally arises when studying the effect of a
subsystem to the eigenmodes of the full system. The proposed method is based on
interpolation of the resolvent function after some of its singularities have
been removed by a spectral projection. Singular value decomposition can be used
to further reduce the dimension of the computational problem. Error analysis of
the method indicates exponential convergence with respect to the number of
interpolation points. Theoretical results are illustrated by two numerical
examples related to finite element discretisation of the Laplace operator
Distributed solution of Laplacian eigenvalue problems
The purpose of this article is to approximately compute the eigenvalues of
the symmetric Dirichlet Laplacian within an interval . A novel
domain decomposition Ritz method, partition of unity condensed pole
interpolation method, is proposed. This method can be used in distributed
computing environments where communication is expensive, e.g., in clusters
running on cloud computing services or networked workstations. The Ritz space
is obtained from local subspaces consistent with a decomposition of the domain
into subdomains. These local subspaces are constructed independently of each
other, using data only related to the corresponding subdomain. Relative
eigenvalue error is analysed. Numerical examples on a cluster of workstations
validate the error analysis and the performance of the method.Comment: 28 page
Efficient solution of symmetric eigenvalue problems from families of coupled systems
Efficient solution of the lowest eigenmodes is studied for a family of related eigenvalue problems with common 2 × 2 block structure. It is assumed that the upper diagonal block varies between different versions while the lower diagonal block and the range of the coupling blocks remain unchanged. Such block structure naturally arises when studying the effect of a subsystem to the eigenmodes of the full system. The proposed method is based on interpolation of the resolvent function after some of its singularities have been removed by a spectral projection. Singular value decomposition can be used to further reduce the dimension of the computational problem. Error analysis of the method indicates exponential convergence with respect to the number of interpolation points. Theoretical results are illustrated by two numerical examples related to finite element discretization of the Laplace operator.Peer reviewe