15,572 research outputs found
Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids
In this paper, we consider anisotropic diffusion with decay, and the
diffusivity coefficient to be a second-order symmetric and positive definite
tensor. It is well-known that this particular equation is a second-order
elliptic equation, and satisfies a maximum principle under certain regularity
assumptions. However, the finite element implementation of the classical
Galerkin formulation for both anisotropic and isotropic diffusion with decay
does not respect the maximum principle.
We first show that the numerical accuracy of the classical Galerkin
formulation deteriorates dramatically with increase in the decay coefficient
for isotropic medium and violates the discrete maximum principle. However, in
the case of isotropic medium, the extent of violation decreases with mesh
refinement. We then show that, in the case of anisotropic medium, the classical
Galerkin formulation for anisotropic diffusion with decay violates the discrete
maximum principle even at lower values of decay coefficient and does not vanish
with mesh refinement. We then present a methodology for enforcing maximum
principles under the classical Galerkin formulation for anisotropic diffusion
with decay on general computational grids using optimization techniques.
Representative numerical results (which take into account anisotropy and
heterogeneity) are presented to illustrate the performance of the proposed
formulation
Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems
We show global well-posedness and exponential stability of equilibria for a
general class of nonlinear dissipative bulk-interface systems. They correspond
to thermodynamically consistent gradient structure models of bulk-interface
interaction. The setting includes nonlinear slow and fast diffusion in the bulk
and nonlinear coupled diffusion on the interface. Additional driving mechanisms
can be included and non-smooth geometries and coefficients are admissible, to
some extent. An important application are volume-surface reaction-diffusion
systems with nonlinear coupled diffusion.Comment: 21 page
``String'' formulation of the Dynamics of the Forward Interest Rate Curve
We propose a formulation of the term structure of interest rates in which the
forward curve is seen as the deformation of a string. We derive the general
condition that the partial differential equations governing the motion of such
string must obey in order to account for the condition of absence of arbitrage
opportunities. This condition takes a form similar to a fluctuation-dissipation
theorem, albeit on the same quantity (the forward rate), linking the bias to
the covariance of variation fluctuations. We provide the general structure of
the models that obey this constraint in the framework of stochastic partial
(possibly non-linear) differential equations. We derive the general solution
for the pricing and hedging of interest rate derivatives within this framework,
albeit for the linear case (we also provide in the appendix a simple and
intuitive derivation of the standard European option problem). We also show how
the ``string'' formulation simplifies into a standard N-factor model under a
Galerkin approximation.Comment: 24 pages, European Physical Journal B (in press
Global stability of steady states in the classical Stefan problem
The classical one-phase Stefan problem (without surface tension) allows for a
continuum of steady state solutions, given by an arbitrary (but sufficiently
smooth) domain together with zero temperature. We prove global-in-time
stability of such steady states, assuming a sufficient degree of smoothness on
the initial domain, but without any a priori restriction on the convexity
properties of the initial shape. This is an extension of our previous result
[28] in which we studied nearly spherical shapes.Comment: 14 pages. arXiv admin note: substantial text overlap with
arXiv:1212.142
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