Friedrichs-type Inequalities Research Articles

Compactness in Banach function spaces: Poincaré and Friedrichs inequalities

In this work, we study the relative compactness of subsets of separable subspaces XsΩ of so-called additive Banach function spaces XΩ, which include the rearrangement-invariant spaces defined on the bounded domain Ω⊂Rn. We choose XsΩ such that the infinitely differentiable functions are dense in it. Moreover, we define the Banach–Sobolev spaces WXsmΩ generated by the above subspaces and we study the compactness of embedding between such spaces. The obtained results are used to establish the equivalent norms on these spaces. These results allow us to prove the Poincaré and Friedrichs-type inequalities for such Sobolev spaces.

Complexes of Discrete Distributional Differential Forms and Their Homology Theory

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus, we generalize a notion of Braess and Schoberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincare–Friedrichs-type inequalities will be studied in a subsequent contribution.

Boundary Value Problem for Stationary Stokes Equations with Impermeability Boundary Condition

We propose two approaches to the study of the boundary value problem for the stationary Stokes equations with impermeability boundary condition. The first approach is classical and is based on a Friedrichs type inequality and a variant of the de Rham theorem. The second approach is based on solving the boundary value problem with the impermeability condition for the system of Poisson equations and decomposition of a Sobolev space into the sum of solenoidal and potential subspaces. We also study the gradient-divergence boundary value problem with impermeability boundary condition and establish the corresponding Ladyzhenskaya–Babushka–Brezzi inequality.

Computable majorants of constants in the Poincaré and Friedrichs inequalities

A method of computing guaranteed bounds of the constants in Poincaré and Friedrichs type inequalities is proposed. This method can be applied to complicated (including multiconnected) domains decomposed into simple nonoverlapping domains. The method can be extended to functions vanishing on a part of the boundary and provides guaranteed bounds of the constant in the Friedrichs inequality. Bibliography: 11 titles. Illustration: 2 figures.

On Friedrichs-type inequalities in domains rarely perforated along the boundary

Abstract This article is devoted to the Friedrichs inequality, where the domain is periodically perforated along the boundary. It is assumed that the functions satisfy homogeneous Neumann boundary conditions on the outer boundary and that they vanish on the perforation. In particular, it is proved that the best constant in the inequality converges to the best constant in a Friedrichs-type inequality as the size of the perforation goes to zero much faster than the period of perforation. The limit Friedrichs-type inequality is valid for functions in the Sobolev space H 1. AMS 2010 Subject Classification: 39A10; 39A11; 39A70; 39B62; 41A44; 45A05.

Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary-value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn- and Friedrichs-type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L2 norm of a vector-valued function from H1(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L2 norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.

A CLASS OF STATIONARY NONLINEAR MAXWELL SYSTEMS

We study a new class of electromagnetostatic problems in the variational framework of the subspace of W1,p(Ω) of vector functions with zero divergence and zero normal trace, for [Formula: see text], in smooth, bounded and simply connected domains Ω of ℝ3. We prove a Poincaré–Friedrichs type inequality and we obtain the existence of steady-state solutions for an electromagnetic induction heating problem and for a quasi-variational inequality modelling a critical state generalized problem for type-II superconductors.

Estimates of deviations from exact solutions of variational inequalities based upon Payne–Weinberger inequality

A new method for obtaining computable estimates for the difference between exact solutions of elliptic variational inequalities and arbitrary functions in the respective energy space is suggested. The estimates are obtained by transforming the corresponding variational inequality without the use of variational duality arguments. These estimates are valid for any function in the energy class and contain no constants depending on the mesh used to find an approximate solution. This method for linear elliptic and parabolic problems was earlier suggested by the author. The guaranteed error bounds we derive can be of two types. Estimates of the first type contain only one global constant, which is a constant in the Friedrichs type inequality. Estimates of the second type are based on the decomposition of Ω into convex subdomains and the Payne-Weinberger inequalities for these subdomains. Bibliography: 20 titles.

Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincare--Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.

On Friedrichs–Poincaré-type inequalities

Friedrichs- and Poincaré-type inequalities are important and widely used in the area of partial differential equations and numerical analysis. Most of their proofs appearing in references are the argument of reduction to absurdity. In this paper, we give direct proofs of Friedrichs-type inequalities in H 1 ( Ω ) and Poincaré-type inequalities in some subspaces of W 1 , p ( Ω ) . The dependencies of the inequality coefficients on the domain Ω and some sub-domains are illustrated explicitly.

Quasioptimality of some spectral mixed methods

In this paper, we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree. Using the projectors, we obtain quasioptimality of some spectral mixed methods, including the Raviart–Thomas method and mixed formulations of Maxwell equations. We also prove some discrete Friedrichs type inequalities involving curl.