56 research outputs found
Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications
We develop a constructive piecewise polynomial approximation theory in
weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The
main ingredients to derive optimal error estimates for an averaged Taylor
polynomial are a suitable weighted Poincare inequality, a cancellation property
and a simple induction argument. We also construct a quasi-interpolation
operator, built on local averages over stars, which is well defined for
functions in . We derive optimal error estimates for any polynomial degree
on simplicial shape regular meshes. On rectangular meshes, these estimates are
valid under the condition that neighboring elements have comparable size, which
yields optimal anisotropic error estimates over -rectangular domains. The
interpolation theory extends to cases when the error and function regularity
require different weights. We conclude with three applications: nonuniform
elliptic boundary value problems, elliptic problems with singular sources, and
fractional powers of elliptic operators
A FEM for an optimal control problem of fractional powers of elliptic operators
We study solution techniques for a linear-quadratic optimal control problem
involving fractional powers of elliptic operators. These fractional operators
can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic
problem posed on a semi-infinite cylinder in one more spatial dimension. Thus,
we consider an equivalent formulation with a nonuniformly elliptic operator as
state equation. The rapid decay of the solution to this problem suggests a
truncation that is suitable for numerical approximation. We discretize the
proposed truncated state equation using first degree tensor product finite
elements on anisotropic meshes. For the control problem we analyze two
approaches: one that is semi-discrete based on the so-called variational
approach, where the control is not discretized, and the other one is fully
discrete via the discretization of the control by piecewise constant functions.
For both approaches, we derive a priori error estimates with respect to the
degrees of freedom. Numerical experiments validate the derived error estimates
and reveal a competitive performance of anisotropic over quasi-uniform
refinement
Positive approximations of the inverse of fractional powers of SPD M-matrices
This study is motivated by the recent development in the fractional calculus
and its applications. During last few years, several different techniques are
proposed to localize the nonlocal fractional diffusion operator. They are based
on transformation of the original problem to a local elliptic or
pseudoparabolic problem, or to an integral representation of the solution, thus
increasing the dimension of the computational domain. More recently, an
alternative approach aimed at reducing the computational complexity was
developed. The linear algebraic system , is considered, where is a properly normalized (scalded) symmetric
and positive definite matrix obtained from finite element or finite difference
approximation of second order elliptic problems in ,
. The method is based on best uniform rational approximations (BURA)
of the function for and natural .
The maximum principles are among the major qualitative properties of linear
elliptic operators/PDEs. In many studies and applications, it is important that
such properties are preserved by the selected numerical solution method. In
this paper we present and analyze the properties of positive approximations of
obtained by the BURA technique. Sufficient conditions for
positiveness are proven, complemented by sharp error estimates. The theoretical
results are supported by representative numerical tests
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