Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain which is decomposed into an overlapping collection of cylindrical subregions of the form , for . Each of the space-time domains are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters and . In particular, the different space-time grids need not match on the regions of overlap, and the time steps can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit -scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument.

Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and heterogeneous hyperbolic-parabolic approximations of parabolic equations.

     

Convergence rates of cascade algorithms

We consider solutions of a refinement equation of the form

where is a finitely supported sequence called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the cascade algorithm associated with , i.e., the convergence of the sequence in the -norm.

Our main result gives estimates for the convergence rate of the cascade algorithm. Let be the normalized solution of the above refinement equation with the dilation matrix being isotropic. Suppose lies in the Lipschitz space , where 0$"> and . Under appropriate conditions on , the following estimate will be established:

where and is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.

     

Transfinite sequences of continuous and Baire class 1 functions

The set of continuous or Baire class 1 functions defined on a metric space is endowed with the natural pointwise partial order. We investigate how the possible lengths of well-ordered monotone sequences (with respect to this order) depend on the space .

     

Maximal monotonicity, conjugation and the duality product

Recently, the authors studied the connection between each maximal monotone operator and a family of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities.

The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.

     

On the inversion of the convolution and Laplace transform

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform of continuous or generalized functions . The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if is continuous, it is in if , and converges in an appropriate norm or Fréchet topology for generalized functions . As a corollary we obtain a new constructive inversion procedure for the convolution transform ; i.e., for given and we construct a sequence of continuous functions such that .

     

ε-optimality criteria for convex programming problems via exact penalty functions

In this paper, we present-optimality criteria for convex programming problems associated with exact penalty functions. Several authors have given various criteria under the assumption that such convex problems and the associated dual problems can be solved. We assume the solvability of neither the convex problem nor the dual problem. To derive our criteria, we estimate the size of the penalty parameter in terms of an-solution for the dual problem.     

Fixed points in the family of convex representations of a maximal monotone operator

Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exists a convex representation of the operator which is a fixed point of this conjugation.

     

SCDA for 3D lattice gases with repulsive interaction

The selfconsistent diagram approximation (SCDA) is generalized for three-dimensional lattice gases with nearest neighbor repulsive interactions. The free energy is represented in a closed form through elementary functions. Thermodynamical (phase diagrams, chemical potential and mean square fluctuations), structural (order parameter, distribution functions) as well as diffusional characteristics are investigated. The calculation results are compared with the Monte Carlo simulation data to demonstrate high precision of the SCDA in reproducing the equilibrium lattice gas characteristics. It is shown that similarly to two-dimensional systems the specific statistical memory effects strongly influence the lattice gas diffusion in the ordered states. Received 7 August 2002 / Received in final form 22 January 2003 Published online 24 April 2003