5 research outputs found
Unconditional stability of semi-implicit discretizations of singular flows
A popular and efficient discretization of evolutions involving the singular
-Laplace operator is based on a factorization of the differential operator
into a linear part which is treated implicitly and a regularized singular
factor which is treated explicitly. It is shown that an unconditional energy
stability property for this semi-implicit time stepping strategy holds. Related
error estimates depend critically on a required regularization parameter.
Numerical experiments reveal reduced experimental convergence rates for smaller
regularization parameters and thereby confirm that this dependence cannot be
avoided in general.Comment: 21 pages, 8 figure
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
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Approximation schemes for materials with discontinuities
Damage and fracture phenomena are related to the evolution of discontinuities both in space and in time. This contribution deals with methods from mathematical and numerical analysis to handle these: Suitable mathematical formulations and time-discrete schemes for problems with discontinuities in time are presented. For the treatment of problems with discontinuities in space, the focus lies on FE-methods for minimization problems in the space of functions of bounded variation. The developed methods are used to introduce fully discrete schemes for a rate-independent damage model and for the viscous approximation of a model for dynamic phase-field fracture. Convergence of the schemes is discussed
Approximation schemes for materials with discontinuities
Damage and fracture phenomena are related to the evolution of discontinuities both in space and in time. This contribution deals with methods from mathematical and numerical analysis to handle these: Suitable mathematical formulations and time-discrete schemes for problems with discontinuities in time are presented. For the treatment of problems with discontinuities in space, the focus lies on FE-methods for minimization problems in the space of functions of bounded variation. The developed methods are used to introduce fully discrete schemes for a rate-independent damage model and for the viscous approximation of a model for dynamic phase-field fracture. Convergence of the schemes is discussed