8,707 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
Anisotropic mesh refinement in polyhedral domains: error estimates with data in L^2(\Omega)
The paper is concerned with the finite element solution of the Poisson
equation with homogeneous Dirichlet boundary condition in a three-dimensional
domain. Anisotropic, graded meshes from a former paper are reused for dealing
with the singular behaviour of the solution in the vicinity of the non-smooth
parts of the boundary. The discretization error is analyzed for the piecewise
linear approximation in the H^1(\Omega)- and L^2(\Omega)-norms by using a new
quasi-interpolation operator. This new interpolant is introduced in order to
prove the estimates for L^2(\Omega)-data in the differential equation which is
not possible for the standard nodal interpolant. These new estimates allow for
the extension of certain error estimates for optimal control problems with
elliptic partial differential equation and for a simpler proof of the discrete
compactness property for edge elements of any order on this kind of finite
element meshes.Comment: 28 pages, 7 figure
Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion
We consider elliptic diffusion problems with a random anisotropic diffusion
coefficient, where, in a notable direction given by a random vector field, the
diffusion strength differs from the diffusion strength perpendicular to this
notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation
of the random vector field and, therefore, also of the solution of the elliptic
diffusion problem. We show that, given regularity of the elliptic diffusion
problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the
regularity of the solution's dependence on the random parameter, also when
considering this higher spatial regularity. This result then implies that
multilevel collocation and multilevel quadrature methods may be used to lessen
the computation complexity when approximating quantities of interest, like the
solution's mean or its second moment, while still yielding the expected rates
of convergence. Numerical examples in three spatial dimensions are provided to
validate the presented theory
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