optimal relaxation parameter for the uzawa method we consider the uzawa method to solve the stationary stokes equations discretized with stable finite elements an iteration step consists of a velocity update un 1 involving the augmented lagrangian operator with 0 followed by the pressure update pn 1 pn div un 1 the so called richardson update we prove that the inf sup constant satisfies 1 and that if 1 1 the iteration converges linearly with a contraction factor 2 1 2 provided 0 2 this yields the optimal value regardless of

control of interface evolution in multiphase fluid flows we consider an optimal control problem for the interface in a two dimensional multiphase fluid problem the minimization functional consists of two parts the l 2 distance to a given density profile and the interfacial length we show existence and derive necessary first order optimality conditions for a corresponding phase field approximation of the perimeter functional an unconditionally stable fully discrete scheme which is based on low order finite elements is proposed and convergence of corresponding iterates to solutions of the limiting optimality conditions for vanishing discretization parameters is shown computational studies are included to validate the model including the phase field approximation interface motion and topological changes as well as to study relative effects due to discretization regularization errors and the relation of both parts of the functional

a stochastic spectral projection method for solving pdes in domains composed by overlapping discs spheres and half spaces a new mesh free stochastic spectral method based on a spectral inversion of the poisson integral formula is suggested we consider the cases of laplace equation and the system of elasticity equations for domains consisting of overlapping discs half planes spheres and half spaces a randomized projection method we developed in sabelfeld and loshchina 2010 21 is used to solve the system of linear equations approximating the integral equations whose phase space is the intersection surfaces of the domains

non homogeneous generalized newtonian fluids we show the existence of weak solutions to the system describing the motion of incompressible non homogeneous generalized newtonian fluids if the extra stress tensor s d possesses p structure with p ge frac 3d 2 d 2 and variable viscosity the limiting process in the equation of motion is justified by a variational argument which is new in this context

a fully discrete bem fem for the exterior stokes problem in the plane we reformulate the johnson nedelec approach for the exterior two dimensional stokes problem taking advantage of the parameterization of the artificial boundary the main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem this one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals we give a complete error analysis of both the galerkin and fully discrete galerkin methods

global existence of weak solutions for a viscous two phase model abstract the purpose of this paper is to explore a viscous two phase liquid gas model relevant for well and pipe flow our approach relies on applying suitable modifications of techniques previously used for studying the single phase isothermal navier stokes equations a main issue is the introduction of a novel two phase variant of the potential energy function needed for obtaining fundamental a priori estimates we derive an existence result for weak solutions in a setting where transition to single phase flow is guaranteed not to occur when the initial state is a true mixture of both phases some numerical examples are also included in order to demonstrate characteristic behavior of solutions in particular we illustrate how two phase flow is genuinely different compared to single phase flow concerning the behavior of an initial mass discontinuity

asymptotic expansions of the error of spline galerkin boundary element methods in this paper we analyse the existence of asymptotic expansions of the error of galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels these expansions are obtained in terms of an interpolation operator and are useful for the application of richardson extrapolation and for obtaining sharper error bounds we also present and analyse a family of fully discrete spline galerkin methods for the solution of the same equations following the analysis of galerkin methods we show the existence of asymptotic expansions of the error

global existence and stability of solutions for reaction diffusion functional differential equations in this article a class of reaction diffusion functional differential equations is investigated the global existence and uniqueness of solutions and the stability of the trivial solution are obtained some applications are also discussed the method proposed in this article is a combination of the iterative analysis the fixed point theorem and the comparison principle

optimal control for multi phase fluid stokes problems optimal control for a system consistent of the viscosity dependent stokes equations coupled with a transport equation for the viscosity is studied motivated by a lack of sufficient regularity of the adjoint equations artificial diffusion is introduced to the transport equation the asymptotic behavior of the regularized system is investigated optimality conditions for the regularized optimal control problems are obtained and again the asymptotic behavior is analyzed the lack of uniqueness of solutions to the underlying system is another source of difficulties for the problem under investigation

locking free finite element methods for poroelasticity we propose a new formulation along with a family of finite element schemes for the approximation of the interaction between fluid motion and linear mechanical response of a porous medium known as biot s consolidation problem the steady state version of the system is recast in terms of displacement pressure and volumetric stress and both continuous and discrete formulations are analyzed as compact perturbations of invertible problems employing a fredholm argument in particular the error estimates are derived independently of the lame constants numerical results indicate the satisfactory performance and competitive accuracy of the introduced methods

sharp interface limit for a stokes allen cahn system we consider the sharp interface limit of a coupled stokes allen cahn system when a parameter epsilon 0 that is proportional to the thickness of the diffuse interface tends to zero in a two dimensional bounded domain for sufficiently small times we prove convergence of the solutions of the stokes allen cahn system to solutions of a sharp interface model where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two phase stokes system with an additional contribution to the stress tensor which describes the capillary stress to this end we construct a suitable approximation of the solution of the stokes allen cahn system using three levels of the terms in the formally matched asymptotic calculations and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized allen cahn operator moreover a careful treatment of the coupling terms is needed

on the fokker planck boltzmann equation we consider the boltzmann equation perturbed by fokker planck type operator to overcome the lack of strong a priori estimates and to define a meaningful collision operator we introduce a notion of renormalized solution which enables us to establish stability results for sequences of solutions and global existence for the cauchy problem with large data the proof of stability and existence combines renormalization with an analysis of a defect measure

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

maximal lp lq regularity for the two phase stokes equations model problems abstract in this paper we prove the generalized resolvent estimate and maximal l p l q regularity of the stokes equation with and without surface tension and gravity in the whole space with flat interface we prove r boundedness of solution operators defined in a sector 0 c 0 arg 0 with 0 2 and 0 0 which combined with the fourier multiplier theorem of s g mihlin and the operator valued fourier multiplier theorem of l weis yields the required generalized resolvent estimate and maximal l p l q regularity at the same time one of the character of the paper is to introduce special function spaces e q r n 0 and e p q 0 r n r cf 1 7 and 1 8 which is necessary to treat the situation that the normal component of velocity fields jumps across the interface such spaces never appear in the study of the stokes equations with other boundary conditions like non slip condition navier slip condition robin condition or pure neumann condition appearing in the study of one phase problem cf desch et al 2001 12 farwig and sohr 1994 13 saal 2003 22 shibata and shimada 2007 23 shibata and shimizu 2008 25 2009 26 in press 27 because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary

an existence theorem for the multifluid navier stokes problem abstract existence of a weak solution of the navier stokes problem describing a multifluid flow is proved the velocity fields associated to each fluid solves the navier stokes equations in a time dependent domain classical immiscibility conditions on the varying fluids interfaces are taken into account by a new formulation of the problem this formulation was introduced in 5 and used in numerical computations this paper follows a previous one where an existence theorem for stokes multifluid problems was derived 7

existence of mild solutions to stochastic neutral partial functional differential equations with non lipschitz coefficients under a non lipschitz condition with the lipschitz condition being considered as a special case and a weakened linear growth condition the existence and uniqueness of mild solutions to stochastic neutral partial functional differential equations snpfdes is investigated some results in govindan 2003 2005 2 6 are generalized to cover a class of more general snpfdes

a bem fem overlapping algorithm for the stokes equation in this work we study the numerical solution of the dirichlet problem for the stokes equation by overlapping boundary and finite elements the domain where we state our problem is the intersection of a polyhedron with the exterior of a strictly contained obstacle with smooth boundary the solution is then decoupled as a sum of an incident flow defined on the polyhedron domain plus the response of the interior obstacle expressed as a single layer potential and constructed only on the exterior of the obstacle the numerical algorithm follows closely this ansatz by replacing the continuous terms of this decomposition by a finite and a boundary element respectively we prove that under not very restrictive assumptions the method is well defined and converges to the exact solution with the same order as the best approximation of the solution by the discrete spaces in the natural norms of the problem finally in the last section we show how this method can be implemented overcoming some of the difficulties appearing in the implementation and demonstrating its applicability to practical problems

a local regularization operator for triangular and quadrilateral finite elements this paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes this operator is a variant of the regularization operator of clement however ours is constructed via a local projection in a reference domain we prove in this paper that it has the same optimal approximation properties as the standard interpolation operator and we present some applications

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions