489 research outputs found
Convergence of regularized time-stepping methods for differential variational inequalities
2013-2014 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe
Notes on a PDE System for Biological Network Formation
We present new analytical and numerical results for the elliptic-parabolic
system of partial differential equations proposed by Hu and Cai, which models
the formation of biological transport networks. The model describes the
pressure field using a Darcy's type equation and the dynamics of the
conductance network under pressure force effects. Randomness in the material
structure is represented by a linear diffusion term and conductance relaxation
by an algebraic decay term. The analytical part extends the results of
Haskovec, Markowich and Perthame regarding the existence of weak and mild
solutions to the whole range of meaningful relaxation exponents. Moreover, we
prove finite time extinction or break-down of solutions in the spatially
onedimensional setting for certain ranges of the relaxation exponent. We also
construct stationary solutions for the case of vanishing diffusion and critical
value of the relaxation exponent, using a variational formulation and a penalty
method. The analytical part is complemented by extensive numerical simulations.
We propose a discretization based on mixed finite elements and study the
qualitative properties of network structures for various parameters values.
Furthermore, we indicate numerically that some analytical results proved for
the spatially one-dimensional setting are likely to be valid also in several
space dimensions.Comment: 33 pages, 12 figure
An a posteriori error estimate for Symplectic Euler approximation of optimal control problems
This work focuses on numerical solutions of optimal control problems. A time
discretization error representation is derived for the approximation of the
associated value function. It concerns Symplectic Euler solutions of the
Hamiltonian system connected with the optimal control problem. The error
representation has a leading order term consisting of an error density that is
computable from Symplectic Euler solutions. Under an assumption of the pathwise
convergence of the approximate dual function as the maximum time step goes to
zero, we prove that the remainder is of higher order than the leading error
density part in the error representation. With the error representation, it is
possible to perform adaptive time stepping. We apply an adaptive algorithm
originally developed for ordinary differential equations. The performance is
illustrated by numerical tests
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
A new class of fractional impulsive differential hemivariational inequalities with an application
We consider a new fractional impulsive differential hemivariational inequality, which captures the required characteristics of both the hemivariational inequality and the fractional impulsive differential equation within the same framework. By utilizing a surjectivity theorem and a fixed point theorem we establish an existence and uniqueness theorem for such a problem. Moreover, we investigate the perturbation problem of the fractional impulsive differential hemivariational inequality to prove a convergence result, which describes the stability of the solution in relation to perturbation data. Finally, our main results are applied to obtain some new results for a frictional contact problem with the surface traction driven by the fractional impulsive differential equation
A rate-independent model for the isothermal quasi-static evolution of shape-memory materials
This note addresses a three-dimensional model for isothermal stress-induced
transformation in shape-memory polycrystalline materials. We treat the problem
within the framework of the energetic formulation of rate-independent processes
and investigate existence and continuous dependence issues at both the
constitutive relation and quasi-static evolution level. Moreover, we focus on
time and space approximation as well as on regularization and parameter
asymptotics.Comment: 33 pages, 3 figure
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