5 research outputs found
Variational Theory and Domain Decomposition for Nonlocal Problems
In this article we present the first results on domain decomposition methods
for nonlocal operators. We present a nonlocal variational formulation for these
operators and establish the well-posedness of associated boundary value
problems, proving a nonlocal Poincar\'{e} inequality. To determine the
conditioning of the discretized operator, we prove a spectral equivalence which
leads to a mesh size independent upper bound for the condition number of the
stiffness matrix. We then introduce a nonlocal two-domain variational
formulation utilizing nonlocal transmission conditions, and prove equivalence
with the single-domain formulation. A nonlocal Schur complement is introduced.
We establish condition number bounds for the nonlocal stiffness and Schur
complement matrices. Supporting numerical experiments demonstrating the
conditioning of the nonlocal one- and two-domain problems are presented.Comment: Updated the technical part. In press in Applied Mathematics and
Computatio
Some error estimates for the finite volume element method for a parabolic problem
We study spatially semidiscrete and fully discrete finite volume element
methods for the homogeneous heat equation with homogeneous Dirichlet boundary
conditions and derive error estimates for smooth and nonsmooth initial data. We
show that the results of our earlier work \cite{clt11} for the lumped mass
method carry over to the present situation. In particular, in order for error
estimates for initial data only in to be of optimal second order for
positive time, a special condition is required, which is satisfied for
symmetric triangulations. Without any such condition, only first order
convergence can be shown, which is illustrated by a counterexample.
Improvements hold for triangulations that are almost symmetric and piecewise
almost symmetric
SOME NEW ERROR ESTIMATES OF A SEMIDISCRETE FINITE VOLUME ELEMENT METHOD FOR A PARABOLIC INTEGRO-DIFFERENTIAL EQUATION WITH NONSMOOTH INITIAL DATA ∗
Abstract. A semidiscrete finite volume element (FVE) approximation to a parabolic integrodifferential equation (PIDE) is analyzed in a two-dimensional convex polygonal domain. An optimalorder L 2-error estimate for smooth initial data and nearly the same optimal-order L 2-error estimate for nonsmooth initial data are obtained. More precisely, for homogeneous equations, an elementary energy technique and a duality argument are used to derive an error estimate of order O � t −1 h 2 ln h � in the L 2-norm for positive time when the given initial function is only in L 2