9,750 research outputs found
Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential
Equation which originated in geometric surface theory, and has been applied in
dynamic meteorology, elasticity, geometric optics, image processing and image
registration. Solutions can be singular, in which case standard numerical
approaches fail. In this article we build a finite difference solver for the
Monge-Ampere equation, which converges even for singular solutions. Regularity
results are used to select a priori between a stable, provably convergent
monotone discretization and an accurate finite difference discretization in
different regions of the computational domain. This allows singular solutions
to be computed using a stable method, and regular solutions to be computed more
accurately. The resulting nonlinear equations are then solved by Newton's
method. Computational results in two and three dimensions validate the claims
of accuracy and solution speed. A computational example is presented which
demonstrates the necessity of the use of the monotone scheme near
singularities.Comment: 23 pages, 4 figures, 4 tables; added arxiv links to references, added
coment
Global existence for a singular phase field system related to a sliding mode control problem
In the present contribution we consider a singular phase field system located
in a smooth and bounded three-dimensional domain. The entropy balance equation
is perturbed by a logarithmic nonlinearity and by the presence of an additional
term involving a possibly nonlocal maximal monotone operator and arising from a
class of sliding mode control problems. The second equation of the system
accounts for the phase dynamics, and it is deduced from a balance law for the
microscopic forces that are responsible for the phase transition process. The
resulting system is highly nonlinear; the main difficulties lie in the
contemporary presence of two nonlinearities, one of which under time
derivative, in the entropy balance equation. Consequently, we are able to prove
only the existence of solutions. To this aim, we will introduce a backward
finite differences scheme and argue on this by proving uniform estimates and
passing to the limit on the time step.Comment: Key words: Phase field system; maximal monotone nonlinearities;
nonlocal terms; initial and boundary value problem; existence of solution
Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions
We prove well-posedness results for the solution to an initial and
boundary-value problem for an Allen-Cahn type equation describing the
phenomenon of phase transitions for a material contained in a bounded and
regular domain. The dynamic boundary conditions for the order parameter have
been recently proposed by some physicists to account for interactions with the
walls. We show our results using suitable regularizations of the nonlinearities
of the problem and performing some a priori estimates which allow us to pass to
the limit thanks to compactness and monotonicity arguments.Comment: Key words: Allen-Cahn equation, dynamic boundary conditions, maximal
monotone graphs, initial boundary value problem, existence and uniqueness
result
The boundary Riemann solver coming from the real vanishing viscosity approximation
We study a family of initial boundary value problems associated to mixed
hyperbolic-parabolic systems:
v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x =
\epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx}
The conservative case is, in particular, included in the previous
formulation.
We suppose that the solutions to these problems converge to a
unique limit. Also, it is assumed smallness of the total variation and other
technical hypotheses and it is provided a complete characterization of the
limit.
The most interesting points are the following two.
First, the boundary characteristic case is considered, i.e. one eigenvalue of
can be .
Second, we take into account the possibility that is not invertible. To
deal with this case, we take as hypotheses conditions that were introduced by
Kawashima and Shizuta relying on physically meaningful examples. We also
introduce a new condition of block linear degeneracy. We prove that, if it is
not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added
Section 3.1.2. Minor changes in other section
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