On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and asymptotic behaviour of solutions for a class of quasi‐linear evolution equations with non‐linear damping and source terms

We consider a class of quasi‐linear evolution equations with non‐linear damping and source terms arising from the models of non‐linear viscoelasticity. By a Galerkin approximation scheme combined with the potential well method we prove that when m<p, where m(?0) and p are, respectively, the growth orders of the non‐linear strain terms and the source term, under appropriate conditions, the initial boundary value problem of the above‐mentioned equations admits global weak solutions and the solutions decay to zero as t→∞. Copyright © 2002 John Wiley & Sons, Ltd.     

WEB‐spline based mesh‐free finite element analysis for the heat equation and the time‐dependent Navier–Stokes equation: A survey

In this article, we give some numerical techniques and error estimates using web‐spline based mesh‐free finite element method for the heat equation and the time‐dependent Navier–Stokes equations on bounded domains. The web‐spline method uses weighted extended B‐splines on a regular grid as basis functions and does not require any grid generation. We demonstrate the method by providing numerical results for the Poisson's and stationary Stokes equation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The hp‐version of the BEM with quasi‐uniform meshes for a three‐dimensional crack problem: The case of a smooth crack having smooth boundary curve

The article considers a three‐dimensional crack problem in linear elasticity with Dirichlet boundary conditions. The crack in this model problem is assumed to be a smooth open surface with smooth boundary curve. The hp‐version of the boundary element method with weakly singular operator is applied to approximate the unknown jump of the traction which is not L₂‐regular due to strong edge singularities. Assuming quasi‐uniform meshes and uniform distributions of polynomial degrees, we prove an a priori error estimate in the energy norm. The estimate gives an upper bound for the error in terms of the mesh size h and the polynomial degree p. It is optimal in h for any given data and quasi‐optimal in p for sufficiently smooth data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Exponential attractor for a planar shear‐thinning flow

We study the dynamics of an incompressible, homogeneous fluid of a power‐law type, with the stress tensor T = ν(1 + µ|Dv|)^p?2Dv, where Dv is a symmetric velocity gradient. We consider the two‐dimensional problem with periodic boundary conditions and p ∈ (1, 2). Under these assumptions, we estimate the fractal dimension of the exponential attractor, using the so‐called method of ??‐trajectories. Copyright © 2007 John Wiley & Sons, Ltd.     

On uniqueness and time regularity of flows of power‐law like non‐Newtonian fluids

Large class of non‐Newtonian fluids can be characterized by index p, which gives the growth of the constitutively determined part of the Cauchy stress tensor. In this paper, the uniqueness and the time regularity of flows of these fluids in an open bounded three‐dimensional domain is established for subcritical ps, i.e. for p>11/5. Our method works for ‘all’ physically relevant boundary conditions, the Cauchy stress need not be potential and it may depend explicitly on spatial and time variable. As a simple consequence of time regularity, pressure can be introduced as an integrable function even for Dirichlet boundary conditions. Moreover, these results allow us to define a dynamical system corresponding to the problem and to establish the existence of an exponential attractor. Copyright © 2010 John Wiley & Sons, Ltd.     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Incompressible non‐Newtonian fluids: time asymptotic behaviour of weak solutions

We study the asymptotic behaviour in time of incompressible non‐Newtonian fluids in the whole space assuming that initial data also belong to L¹. Firstly, we consider the weak solution to the power‐law model with non‐zero external forces and we find the asymptotic behaviour in time of this solution in the same class of existence and uniqueness with p?. Secondly, we are interested in the asymptotic behaviour of weak solutions to the second grade model, and finally, we deal with the asymptotic behaviour in time of weak solutions to a simplified model of viscoelastic fluids of the Oldroyd type. Copyright © 2006 John Wiley & Sons, Ltd.     

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi‐Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi‐Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first‐order convergence in the discrete L²‐norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes.     

A mixed finite element method for a quasi‐Newtonian fluid flow

We propose a mixed formulation for quasi‐Newtonian fluid flow obeying the power law where the stress tensor is introduced as a new variable. Based on such a formulation, a mixed finite element is constructed and analyzed. This finite element method possesses local (i.e., at element level) conservation properties (conservation of the momentum and the mass) as in the finite volume methods. We give existence and uniqueness results for the continuous problem and its approximation and we prove error bounds. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

On the best constant in a Wente‐type inequality for the fractional Laplace operator

In this paper, we consider the solution to Wente's problem with the fractional Laplace operator (?Δ)^α/2, where 0 < α < 2. We derive a Wente‐type inequality for this problem. Next, we compute the optimal constant in such inequality. Copyright © 2015 John Wiley & Sons, Ltd.     

Web‐compact spaces,Fréchet‐Urysohn groups and a Suslin closed graph theorem

A topological space X is strongly web‐compact if X admits a family {A_α: α ∈ ?^?} of relatively countably compact sets covering X and such that A_α ? A_β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet‐Urysohn and Baire and Y is strongly web‐compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web‐compact is not productive. This applies to show that there are quasi‐Suslin spaces X whose product X × X is not quasi‐Suslin (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of nonconstant steady‐state solutions for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor

We consider the periodic problem for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor. By choosing a suitable symmetrizers and using an induction argument on the order of the time‐space derivatives of solutions in energy estimates, we obtain the global stability of solutions with exponential decay in time near the nonconstant steady‐states for 2‐fluid nonisentropic Euler‐Poisson equations. This improves the results obtained for models with temperature diffusion terms by using the pressure functions p^ν in place of the unknown variables densities n^ν.     

On the evolution of sharp fronts for the quasi‐geostrophic equation

We consider the problem of the evolution of sharp fronts for the surface quasi‐geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two‐dimensional Euler equation. The special interest of the quasi‐geostrophic equation lies in its strong similarities with the three‐dimensional Euler equation, while being a two‐dimen‐sional model. In particular, an analogue of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for the three‐dimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot‐Savart law still needs to be understood. We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve. Finally, using a Nash‐Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C^∞ case. © 2004 Wiley Periodicals, Inc.     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

Locally boundedly A‐p‐convex algebras

We consider locally p‐convex algebras the bornologies of which coincide with the ones of p‐normed or multiplicatively p‐convex topologies. Two new classes of these algebras are introduced here. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the well‐posedness of a mathematical model describing water‐mud interaction

In this paper, we consider a mathematical model describing the two‐phase interaction between water and mud in a water canal when the width of the canal is small compared with its depth. The mud is treated as a non‐Newtonian fluid, and the interface between the mud and fluid is allowed to move under the influence of gravity and surface tension. We reduce the mathematical formulation, for small boundary and initial data, to a fully nonlocal and nonlinear problem and prove its local well‐posedness by using abstract parabolic theory. Copyright © 2012 John Wiley & Sons, Ltd.     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

Highlights are the following:

• For any integer , we construct ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions.
• To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly.
• To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions.

Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains.