Coarse space with Neumann boundary conditions
. In order to improve the coarse space and, in particular, to be able to prove an estimate for the condition number, we introduce some slight but significant modifications to the method.
In this paper, we will investigate the finite volume element method for the general elliptic optimal control problem with Dirichlet or Neumann boundary conditions
. The variational discretization approach is used to deal with the control, which can avoid explicit discretization of the control and improve the approximation.
We consider interior Neumann functions with several different Neumann boundary conditions
We consider the following equation with Neumann boundary conditions
This corresponds to what is called Neumann boundary conditions
in the literature; see .
Section 4 presents the solutions obtained by the proposed method as well as an error comparison with Mathematica's NDSolve for the one- and two-dimensional cases of our model as well as both Dirichlet and Neumann boundary conditions
. In Section 5 we provide a real-world application of this method and model in the form of document image binarization, illustrating the ability to obtain a useful result with the present method when coupled with the finite-difference discretization.
Li, "Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions
," Journal of Mathematical Analysis and Applications, vol.
where B = 1 represents the Dirichlet boundary condition; B = [partial derivative]/[partial derivative]n represents the Neumann boundary condition
. Applying Dirichlet and Neumann boundary conditions
, we obtained
The source type nonlinear low terms and Neumann boundary conditions
can not make us use Poincare type inequality directly, thanks to strong absorptive terms -[[phi].sup.3] and -[[phi].sup.3], which guarantees the existence of a global solution and will not blow up.
This type of von Neumann or Fourier stability analysis is directly valid for pure initial-value problems and for initial-boundary-value problems with periodic boundary conditions, and it can also be applied to general Dirichlet or Neumann boundary conditions
by using appropriate periodic extensions, provided that consistent discretizations are used for the boundary conditions [21, Ch.
For only Neumann boundary conditions
, with substitutions [[omega].sub.1] = 1, [[omega].sub.2] = [omega], the solution structure can be written in this form:
For non-linear systems, we usually derive a very good approximation to the solutions with the Neumann boundary conditions
. It is also important that the Adomian decomposition method does not require discretization of the variables.