Differential equations with discontinuous right-hand sides☆

Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

http://materias.fi.uba.ar/7538/material/Otros/EymardEtAl%20-%20Finite%20Volume%20Methods.pdf

Finite Volume Methods

Cone metric spaces and fixed point theorems of contractive mappings

A criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant measures for the associated Markov semigroup. The results are applied to reation diffusion equations with additive white noise and to Navier-Stokes equations with multiplicative and with additive white noise.

Attractors for random dynamical systems

This paper presents an abstract analysis of bounded variation (BV) methods for ill-posed operator equations Au=z Let T(u)def=//Au-z//2+ alpha J(u) where the penalty, or 'regularization parameter alpha >0 and the functional J(u) is the BV norm or semi-norm of u, also known as the total variation of u Under mild restrictions on the operator A and the functional J(u), it is shown that the functional T(u) has a unique minimizer which is stable with respect to certain perturbations in the data z, the operator A, the parameter alpha , and the functional J(u) In addition, convergence results are obtained which apply when these perturbations vanish and the regularization parameter is chosen appropriately

https://iopscience.iop.org/article/10.1088/0266-5611/10/6/003/pdf

Analysis of bounded variation penalty methods for ill-posed problems

This graduate-level text offers a survey of the main ideas, concepts, and methods that constitute nonlinear functional analysis. It features extensive commentary, many examples, and interesting, challenging exercises. Topics include degree mappings for infinite dimensional spaces, the inverse function theory, the implicit function theory, Newton's methods, and many other subjects. 1985 edition.

Klaus Deimling

Papers

Nonlinear Functional Analysis