1,944 research outputs found
Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator
This paper introduces a novel method for the efficient and accurate
computation of the volume of a domain whose boundary is given by an orientable
hypersurface which is implicitly given as the iso-contour of a sufficiently
smooth level-set function. After spatial discretization, local approximation of
the hypersurface and application of the Gaussian divergence theorem, the volume
integrals are transformed to surface integrals. Application of the surface
divergence theorem allows for a further reduction to line integrals which are
advantageous for numerical quadrature. We discuss the theoretical foundations
and provide details of the numerical algorithm. Finally, we present numerical
results for convex and non-convex hypersurfaces embedded in cuboidal domains,
showing both high accuracy and thrid- to fourth-order convergence in space.Comment: 25 pages, 17 figures, 3 table
On the convergence of second order spectra and multiplicity
Let A be a self-adjoint operator acting on a Hilbert space. The notion of
second order spectrum of A relative to a given finite-dimensional subspace L
has been studied recently in connection with the phenomenon of spectral
pollution in the Galerkin method. We establish in this paper a general
framework allowing us to determine how the second order spectrum encodes
precise information about the multiplicity of the isolated eigenvalues of A.
Our theoretical findings are supported by various numerical experiments on the
computation of inclusions for eigenvalues of benchmark differential operators
via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
A RBF partition of unity collocation method based on finite difference for initial-boundary value problems
Meshfree radial basis function (RBF) methods are popular tools used to
numerically solve partial differential equations (PDEs). They take advantage of
being flexible with respect to geometry, easy to implement in higher
dimensions, and can also provide high order convergence. Since one of the main
disadvantages of global RBF-based methods is generally the computational cost
associated with the solution of large linear systems, in this paper we focus on
a localizing RBF partition of unity method (RBF-PUM) based on a finite
difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation
method, which can successfully be applied to solve time-dependent PDEs. This
approach allows to significantly decrease ill-conditioning of traditional
RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix
system, reducing the computational effort but maintaining at the same time a
high level of accuracy. Numerical experiments show performances of our
collocation scheme on two benchmark problems, involving unsteady
convection-diffusion and pseudo-parabolic equations
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