Friedrichs Inequality Research Articles

On the linear convergence of additive Schwarz methods for the p-Laplacian

Abstract We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. First, we present a new convergence theory for additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behaviour akin to the Bregman distance of the convex energy functional associated with the problem. Secondly, we provide a quasi-norm version of the Poincaré–Friedrichs inequality, which plays a crucial role in deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these key elements we establish the asymptotic linear convergence of additive Schwarz methods for the $p$-Laplacian.

Representation formulae for the higher-order Steklov and $L^{2^m}$-Friedrichs inequalities

Representation formulae for the higher-order Steklov and $L^{2^m}$-Friedrichs inequalities

Spectrum of the Dirichlet Laplacian in a thin cubic lattice

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width ε ≪ 1) which have a square cross section. This spectrum coincides with the union of segments which all go to +∞ as ε tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length O(e−δ/ε), δ > 0, while the length of the next spectral segments is O(ε). To establish these results, we need to study in detail the properties of the Dirichlet Laplacian AΩ in the geometry Ω obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max–min arguments as well as a well-chosen Poincaré–Friedrichs inequality, we prove that AΩ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for AΩ, that is no non trivial bounded solution at the threshold frequency for AΩ. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.

Improved Friedrichs inequality for a subhomogeneous embedding

For a smooth bounded domain Ω and p⩾q⩾2, we establish quantified versions of the classical Friedrichs inequality ‖∇u‖pp−λ1‖u‖qp⩾0, u∈W01,p(Ω), where λ1 is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation −Δpu=λ1‖u‖qp−q|u|q−2u+f coupled with zero Dirichlet boundary conditions possesses a weak solution provided f is orthogonal to the minimizer of λ1.

Friedrichs Inequalities and Sharpened Sufficient Stability Conditions of Plane-Parallel Flows

Friedrichs Inequalities and Sharpened Sufficient Stability Conditions of Plane-Parallel Flows

Embedding theorems related to torsional rigidity and principal frequency

We study criteria for the finiteness of the constants in integral inequalities generalizing the Poincaré–Friedrichs inequality and Saint-Venant’s variational definition of torsional rigidity. The Rayleigh–Faber–Krahn isoperimetric inequality and the Saint-Venant–Pólya inequality guarantee the existence of finite constants for domains of finite volume. Criteria for the existence of finite constants for unbounded domains of infinite volume were known only in the cases of planar simply connected and spatial convex domains. We generalize and strengthen some known results and extend them to the case when . Here is one of our results. Suppose that and , where is a compact set and is either a planar domain with uniformly perfect boundary or a spatial domain satisfying the exterior sphere condition. Under these assumptions, a finite constant exists if and only if the integral is finite, where is the distance from the point to the boundary of .

Selected results and open problems on Hardy–Rellich and Poincaré–Friedrichs inequalities

Selected results and open problems on Hardy–Rellich and Poincaré–Friedrichs inequalities

Staggered DG Method for Coupling of the Stokes and Darcy--Forchheimer Problems

In this paper we develop a staggered discontinuous Galerkin method for the Stokes and Darcy--Forchheimer problems coupled with the Beavers--Joseph--Saffman conditions. The method is defined by imposing staggered continuity for all the variables involved and the interface conditions are enforced by switching the roles of the variables met on the interface, which eliminate the hassle of introducing additional variables. This method can be flexibly applied to rough grids such as the highly distorted grids and the polygonal grids. In addition, the method allows nonmatching grids on the interface thanks to the special inclusion of the interface conditions, which is highly appreciated from a practical point of view. A new discrete trace inequality and a generalized Poincaré--Friedrichs inequality are proved, which enables us to prove the optimal convergence estimates under reasonable regularity assumptions. Finally, several numerical experiments are given to illustrate the performances of the proposed method, and the numerical results indicate that the proposed method is accurate and efficient, and in addition, it is a good candidate for practical applications.

Multiple States in Turbulent Large-Aspect-Ratio Thermal Convection: What Determines the Number of Convection Rolls?

Wall-bounded turbulent flows can take different statistically stationary turbulent states, with different transport properties, even for the very same values of the control parameters. What state the system takes depends on the initial conditions. Here we analyze the multiple states in large-aspect ratio (Γ) two-dimensional turbulent Rayleigh-Bénard flow with no-slip plates and horizontally periodic boundary conditions as model system. We determine the number n of convection rolls, their mean aspect ratios Γ_{r}=Γ/n, and the corresponding transport properties of the flow (i.e., the Nusselt number Nu), as function of the control parameters Rayleigh (Ra) and Prandtl number. The effective scaling exponent β in Nu∼Ra^{β} is found to depend on the realized state and thus Γ_{r}, with a larger value for the smaller Γ_{r}. By making use of a generalized Friedrichs inequality, we show that the elliptical shape of the rolls and viscous damping determine the Γ_{r} window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding 2/3≤Γ_{r}≤4/3, where the lower threshold is approached for the larger Ra. Finally, we show that the theoretical approach to frame Γ_{r} also works for free-slip boundary conditions.

Functional analysis and exterior calculus on mixed-dimensional geometries

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d + 1-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.

Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))

This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.

Poincaré–Friedrichs inequalities of complexes of discrete distributional differential forms

We derive bounds for the constants in Poincare–Friedrichs inequalities with respect to mesh-dependent norms for complexes of discrete distributional differential forms. A key tool is a generalized flux reconstruction which is of independent interest. The results apply to piecewise polynomial de Rham sequences on bounded domains with mixed boundary conditions.

Friedrichs inequality in irregular domains

We prove a generalized version of Friedrichs and Gaffney inequalities for a bounded (ε,δ) domain Ω⊂Rn, n=2,3, by adapting the methods of Jones to our framework.

On $\varepsilon$-Friedrichs inequalities and its application

On $\varepsilon$-Friedrichs inequalities and its application

Gaffney–Friedrichs inequality for differential forms on Heisenberg groups

In this paper, we will prove several generalized versions, dependent on different boundary conditions, of the classical Gaffney–Friedrichs inequality for differential forms on Heisenberg groups. In the first part of the paper, we will consider horizontal differential forms and the horizontal differential. In the second part, we shall prove the counterpart of these results in the context of Rumin’s complex.

Stability of an Unsteady Shear of Bingham Medium in the Plane Layer

This work studies the plane-parallel unsteady shear of a homogeneous two-constant viscoplastic Bingham medium in the infinite layer. It was assumed that the axial velocity of the flow as a function of one spatial coordinate and time is known from the solution to the classical one-dimensional unsteady problem. The time variation in the thicknesses of the possible rigid zones is considered; their boundaries are parallel to the boundaries of the layer. The two-dimensional plane perturbations are superposed upon the main flow. The problem in terms of perturbations reduces to one linearized equation for the amplitude of the function of the flow with the corresponding set of four boundary conditions, and several variants of such quadruples are studied here. With the method of integral relationships, the problem reduces to the minimization problem of ratios of quadratic functionals depending on time in the space H2(a;b), where a and b are functions of time determined by the motion of the rigid zones in the main flow. For different variants of imposition of the boundary conditions, the generalized Friedrichs inequalities are proved, and the sufficient integral estimates of stability are derived in which the Reynolds and Saint-Venant numbers and the maximum shear velocity in thickness in the main flow play a role. The dependence of the obtained estimates on the viscous and plastic properties of the medium is discussed.

Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation

Abstract This paper provides a discrete Poincaré inequality in n space dimensions on a simplex K with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on K and all integrals of jumps zero along all interior sides by its Lebesgue norm times C ⁢ ( n ) ⁢ diam ⁡ ( K ) {C(n)\operatorname{diam}(K)} . The explicit constant C ⁢ ( n ) {C(n)} depends only on the dimension n = 2 , 3 {n=2,3} in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle ω 0 {\omega_{0}} in the triangulation. The analysis allows the bound of two constants Λ 1 {\Lambda_{1}} and Λ 3 {\Lambda_{3}} in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.

Qualitative Properties of the Solution to Brinkman-Stokes System Modelling a Filtration Process

The paper deals with study of Stokes-Brinkman system with varying viscosity that describes the fluid flow along an ensemble of partially porous cylindrical particles using the cell approach. Existence and uniqueness of the solution to the system is proved for an arbitrary varying viscosity. Some uniform estimates on the velocity of flow are derived. Moreover, an axillary weighted Friedrichs inequality was proved for the solution of the considered system. The numerical illustration of the obtained results is given.

A Friedrichs–Maz'ya inequality for functions of bounded variation

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Nevertheless, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects.As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with σ‐finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp.

Lower bounds for the eigenvalues of the Spinc Dirac operator on manifolds with boundary

We extend the Friedrich inequality for the eigenvalues of the Dirac operator on Spinc manifolds with boundary under different boundary conditions. The limiting case is then studied and examples are given.