3,438 research outputs found
Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics
In this paper an abstract evolutionary hemivariational inequality with a
history-dependent operator is studied. First, a result on its unique
solvability and solution regularity is proved by applying the Rothe method.
Next, we introduce a numerical scheme to solve the inequality and derive error
estimates. We apply the results to a quasistatic frictional contact problem in
which the material is modeled with a viscoelastic constitutive law, the contact
is given in the form of multivalued normal compliance, and friction is
described with a subgradient of a locally Lipschitz potential. Finally, for the
contact problem we provide the optimal error estimate
Evolutionary Oseen model for generalized Newtonian fluid with Multivalued Nonmonotone Friction Law
The paper deals with the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and nonmonotone frictional slip boundary conditions. First, we provide a result on existence of a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function. We employ a method based on a surjectivity theorem for multivalued L-pseudomonotone operators. Then, we exploit the abstract result to prove the weak unique solvability of the Oseen system
Lecture Notes on Gradient Flows and Optimal Transport
We present a short overview on the strongest variational formulation for
gradient flows of geodesically -convex functionals in metric spaces,
with applications to diffusion equations in Wasserstein spaces of probability
measures. These notes are based on a series of lectures given by the second
author for the Summer School "Optimal transportation: Theory and applications"
in Grenoble during the week of June 22-26, 2009
Stampacchia's property, self-duality and orthogonality relations
We show that if the conclusion of the well known Stampacchia Theorem, on
variational inequalities, holds on a Banach space X, then X is isomorphic to a
Hilbert space. Motivated by this we obtain a relevant result concerning
self-dual Banach spaces and investigate some connections between existing
notions of orthogonality and self-duality. Moreover, we revisit the notion of
the cosine of a linear operator and show that it can be used to characterize
Hilbert space structure. Finally, we present some consequences of our results
to quadratic forms and to evolution triples
Convergence of Rothe scheme for hemivariational inequalities of parabolic type
This article presents the convergence analysis of a sequence of piecewise
constant and piecewise linear functions obtained by the Rothe method to the
solution of the first order evolution partial differential inclusion
, where the multivalued term
is given by the Clarke subdifferential of a locally Lipschitz functional. The
method provides the proof of existence of solutions alternative to the ones
known in literature and together with any method for underlying elliptic
problem, can serve as the effective tool to approximate the solution
numerically. Presented approach puts into the unified framework known results
for multivalued nonmonotone source term and boundary conditions, and
generalizes them to the case where the multivalued term is defined on the
arbitrary reflexive Banach space as long as appropriate conditions are
satisfied. In addition the results on improved convergence as well as the
numerical examples are presented.Comment: to appear in: International Journal of Numerical Analysis and
Modelin
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