a node centered finite volume method for a fracture model on triangulations abstract in this paper a node centered finite volume method based on triangulations for a fracture model is presented in which we restrict the pressure to the linear finite element space while the velocity can be approximated by constant vectors element by element the numerical scheme is established just associated with the pressure to avoid the saddle point problem error estimates of o h accuracy for the discrete h1 semi norm and the discrete l2 norm of pressure p and the l2 2 norm of velocity u are developed on general triangulations under an additional assumption about essentially symmetric control volumes the error estimates for the pressure p can be improved to o h3 2 finally numerical experiments are carried out to verify the accuracy and convergence rates for the proposed finite volume scheme

modeling fractures and barriers as interfaces for flow in porous media we consider a fractured porous medium that is studied at a scale such that the fractures can be modeled individually models for flow in which the fractures are interfaces between subdomains are presented these models take into account interactions between the fractures and the surrounding porous medium existence and uniqueness of the solution to the model problem are proved error estimates show convergence as mathcal o max h d in the l2 norm where h is the mesh size and d the fracture width numerical experiments confirm the theoretical results

spectral triples for the sierpinski gasket abstract we construct a family of spectral triples for the sierpinski gasket k for suitable values of the parameters we determine the dimensional spectrum and recover the hausdorff measure of k in terms of the residue of the volume functional a tr a d s at its abscissa of convergence d d which coincides with the hausdorff dimension d h of the fractal we determine the associated connes distance showing that it is bi lipschitz equivalent to the distance on k induced by the euclidean metric of the plane and show that the pairing of the associated fredholm module with odd k theory is non trivial when the parameters belong to a suitable range the abscissa of convergence d of the energy functional a tr d s 2 d a 2 d s 2 takes the value d e log 12 5 log 2 which we call energy dimension and the corresponding residue gives the standard dirichlet form on k

the blow up rate for a system of heat equations with nonlinear boundary conditions this paper deals with the blow up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions the upper and lower bounds of blow up rate are obtained

partitioning strategies for the interaction of a fluid with a poroelastic material based on a nitsche s coupling approach abstract we develop a computational model to study the interaction of a fluid with a poroelastic material the coupling of stokes and biot equations represents a prototype problem for these phenomena which feature multiple facets on one hand it shares common traits with fluid structure interaction on the other hand it resembles the stokes darcy coupling for these reasons the numerical simulation of the stokes biot coupled system is a challenging task the need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners for the equations at hand are typical shortcomings of classical discretization methods applied to this problem such as the finite element method for spatial discretization and finite differences for time stepping the application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others at each time step in this work we develop and thoroughly analyze a loosely coupled scheme for stokes biot equations the scheme is based on nitsche s method for enforcing interface conditions once the interface operators corresponding to the interface conditions have been defined time lagging allows us to build up a loosely coupled scheme with good stability properties the stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem the error of the resulting method is also analyzed showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes in order to restore good approximation properties while maintaining the computational efficiency of the loosely coupled approach we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

fast fourier transforms for direct solution of poisson s equation with staggered boundary conditions abstract this paper describes pre and postprocessing algorithms used to incorporate the fast fourier transform fft into the solution of finite difference approximations to multi dimensional poisson s equation on a staggered grid where the boundary is located midway between two grid points all frequently occurring boundary conditions neumann dirichlet or cyclic are considered including the combination of staggered neumann boundary condition on one side with nonstaggered dirichlet boundary condition on the other side experiences from implementing these algorithms in vectorized coding in fortran subroutines are reported

numerical treatment of two dimensional interfaces for acoustic and elastic waves we present a numerical method to take into account 2d arbitrary shaped interfaces in classical finite difference schemes on a uniform cartesian grid this work extends the explicit simplified interface method esim previously proposed in 1d j comput phys 168 2001 227 248 the physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics with stationary interfaces our method maintains near the interfaces properties of the schemes in homogeneous medium such as the order of accuracy and the stability limit moreover it enforces the numerical solution to satisfy the exact interface conditions lastly it provides subcell geometrical features of the interface inside the meshing the esim can be coupled automatically with a wide class of numerical schemes lax wendroff flux limiter schemes etc for a negligible additional computational cost throughout the paper we focus on the challenging case of an interface between a fluid and an elastic solid in numerical experiments we provide comparisons between numerical solutions and original analytic solutions showing the efficiency of the method

a block centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system abstract a block centered finite difference method is proposed for solving an unsteady asymptotic coupled model in which the flow is governed by darcy s law both in the one dimensional fracture and two dimensional porous media the second order error estimates in discrete norms are derived on nonuniform rectangular grids for both pressure and velocity the numerical scheme can be extended to nonmatching spatial and temporal grids without loss of accuracy numerical experiments are performed to verify the efficiency and accuracy of the proposed method it is shown that the pressure and velocity are discontinuous across the fracture interface and the fracture indeed acts as the fast pathway or geological barrier in the aquifer system

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

solution of elliptic equations using fast poisson solvers abstract certain nonseparable elliptic equations may be transformed into a sequence of poisson equations the solutions of these equations are efficiently found using fast poisson solvers the method is illustrated by solving for the internal electric potential in convecting plasma clouds

a blw inequality for vector bundles and applications to spectral bounds abstract we prove a non commutative version of the bochner lichnerowicz weitzenbock formula for the bochner laplacian acting on sections of a c algebra bundle or of a real euclidean bundle provided with a suitable connection we then use it to obtain a lower bound on the spectrum in terms of uniform or lp bounds on the curvature and current of the connection

the convergence of spectral and finite difference methods for initial boundary value problems the general theory of compatibility conditions for the differentiability of solutions to initial boundary value problems is well known this paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of high order methods explicit application of boundary conditions bcs that are independent of the initial condition ic results in the compatibility conditions not being satisfied since this is the case in most science and engineering applications it is shown that not only does the error in a spectral method as measured in the maximum norm converge algebraically but the accuracy of finite differences is also reduced for the heat equation with a parabolic ic and dirichlet bcs we prove that the fourier method converges quadratically in the neighborhood of t 0 and the boundaries and quartically for large t when the first order compatibility conditions are violated for the same problem the chebyshev method initially yields quartic convergence and exponential convergence for t 0 in contrast the wave equation subject to the same conditions results in inferior convergence rates with all spectral methods yielding quadratic convergence for all t these results naturally direct attention to finite difference methods that are also algebraically convergent in the case of the wave equation we prove that a second order finite difference method is reduced to 4 3 order convergence and numerically show that a fourth order finite difference scheme is apparently reduced to 3 2 order finally for the wave equation subject to general ics and zero bcs we give a conjecture on the error for a second order finite difference scheme showing that an o n 2log n convergence is possible

semi groupe de la chaleur transverse sur lac algebre d un feuilletage riemannien resume la construction d une connexion transverse sur le fibre des demi densites longitudinales d un feuilletage riemannien permet de presenter sur lac algebre d un feuilletage un calcul differentiel transverse a l ordre 1 c est a dire un gradient transverse a valeurs dans un bimodule tangent a lac algebre puis a partir de la forme de dirichlet et du laplacien associes le semi groupe de la chaleur transverse qui est a priori un semi groupe de contractions completement positives normales de l algebre de von neumann du feuilletage et dont on montre au moins lorsque la variete est compacte qu il est markovien et qu il preserve lac algebre a foliated structure on a manifold gives rise to a canonical connexion on the longitudinal half densities bundle through which a differential calculus is built on the foliationc algebra in the case of a riemannian foliation we get a transverse gradient and the transverse laplace beltrami operator associated with it which generates a quantum semigroup of completely positive contractions this semigroup is shown to be markovian and preserving thec algebra of the foliation

generalizations of two inequalities involving hermitian forms let 1 and n be respectively the greatest and smallest eigenvalues of an n n hermitian matrix h hij and x x1 x2 xn with x x 1 then it is known that 1 1 x hx n and 2 if in addition h is positive definite 1 n 24 1 n x hx x h 1x 1 assuming that y y1 y2 yn and yi 1 i 1 2 n it is shown in this paper that these inequalities remain true if h and h 1 are respectively replaced by the hadamard products m y h and m y h 1 where m y is a matrix defined by m y ij 1 ij yiyj subsequently these results are extended to improve the spectral bounds of m y h

a vorticity based fully mixed formulation for the 3d brinkman darcy problem abstract we propose and analyze a fully mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium an array of low concentration fixed particles described by brinkman equations and its interaction with non viscous flow within classical porous media governed by darcy s law the system is formulated in terms of velocity and pressure in the porous medium together with vorticity velocity and pressure of the viscous fluid in addition and for sake of the analysis the tangential component of the vorticity is supposed to vanish on the whole boundary of the brinkman domain whereas null normal components of both velocities are assumed on the respective boundaries except on the interface where suitable transmission conditions are considered in this way the derivation of the corresponding mixed variational formulation leads to a lagrange multiplier enforcing the pressure continuity across the interface whereas mass balance results from essential boundary conditions on each domain as a consequence a typical saddle point operator equation is obtained and hence the classical babuska brezzi theory is applied to establish the well posedness of the continuous and discrete schemes in particular we remark that the continuous and discrete inf sup conditions of the main bilinear form are proved by using suitably chosen injective operators to get lower bounds of the corresponding suprema which constitutes a previously known technique recently denominated t coercivity in turn and consistently with the above the stability of the galerkin scheme requires that the curl of the finite element subspace approximating the vorticity be contained in the space where the discrete velocity of the fluid lives which yields raviart thomas and nedelec finite element subspaces as feasible choices then we show that the aforementioned constraint can be avoided by augmenting the mixed formulation with a residual arising from the brinkman momentum equation finally several numerical examples illustrating the satisfactory performance of the methods and confirming the theoretical rates of convergence are reported

dirac operators and spectral triples for some fractal sets built on curves abstract we construct spectral triples and in particular dirac operators for the algebra of continuous functions on certain compact metric spaces the triples are countable sums of triples where each summand is based on a curve in the space several fractals like a finitely summable infinite tree and the sierpinski gasket fit naturally within our framework in these cases we show that our spectral triples do describe the geodesic distance and the minkowski dimension as well as more generally the complex fractal dimensions of the space furthermore in the case of the sierpinski gasket the associated dixmier type trace coincides with the normalized hausdorff measure of dimension log 3 log 2

a practical choice of parameters in improved sor newton method with orderings abstract in this paper on the basis of the results of ishihara et al 1997 we first discuss global convergence theorems for the improved sor newton and block sor newton methods with orderings applied to a system of mildly nonlinear equations which includes as a special case the discretized version of the dirichlet problem for the equation u p x ux q y uy f x y u where f is continuously differentiable and fu x y u 0 moreover we propose a practical choice of the multiple relaxation parameters i for them numerical examples are also given

blow up analysis for a system of heat equations with nonlinear flux which obey different laws abstract we consider a system of heat equations u t u and v t v in 0 t completely coupled by nonlinear boundary conditions u e p v u v u q e v on 0 t we prove that the solutions always blow up in finite time for non zero and non negative initial values also the blow up only occurs on with c 1 t t p 2 p q max u x t c 2 t t p 2 p q log c 3 t t q 1 2 p q max v x t log c 4 t t q 1 2 p q for p q 0 0 1 and 0 p

determination of relaxation factors for high cell peclet number flow simulation abstract for the numerical simulation of transport problems with high cell peclet number the coefficient matrix of finite difference equations may lose the diagonal dominance if a scheme more accurate than the first order upwind is used to approximate the convection terms hence in many cases it is difficult to choose a suitable relaxation factor a priori for these schemes when the commonly used iterative method is applied to obtain the convergent solution however it is found that after appropriate normalization an easier determined relaxation factor useful to all three schemes studied here i e second order central differencing second order upwinding and quick exists two model problems are used to compare the performances among the three schemes the second order central differencing is found to be less efficient in the cases investigated without the aid of the normalization the second order upwind scheme has the widest range of allowable values for the relaxation factor yet quick may be computationally more efficient after normalization this is mainly because the number of iterations needed for quick is less sensitive to the choice of the relaxation factor when the optimum value isn t known for a flow problem with non constant velocities the present method becomes the iterative algorithm with different relaxation parameters for different points i e a local relaxation method it is demonstrated that in addition to helping choose the relaxation factors this idea may also substantially reduce the computing time compared with the iterative algorithm with the uniform relaxation factor for all the grid points

measures from dixmier traces and zeta functions for l functions on a closed compact riemannian manifold the noncommutative residue and the dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the lebesgue integral the identifications are shown to continue to and be sharp at l2 functions for functions strictly in lp 1 p 2 symmetrised noncommutative residue and dixmier trace formulas must be introduced for which the identification is shown to continue for the noncommutative residue however a failure is shown for the dixmier trace formulation at l1 functions it is shown the noncommutative residue remains finite and recovers the lebesgue integral for any integrable function while the dixmier trace expression can diverge the results show that a claim in the monograph j m gracia bondia j c varilly h figueroa elements of noncommutative geometry birkhauser adv texts birkhauser boston 2001 that the equality on c functions between the lebesgue integral and an operator theoretic expression involving a dixmier trace obtained from connes trace theorem can be extended to any integrable function is false the results of this paper include a general presentation for finitely generated von neumann algebras of commuting bounded operators including a bounded borel or l functional calculus version of c results in iv 2 of a connes noncommutative geometry academic press new york 1994

dirichlet forms and markovian semigroups on standard forms of von neumann algebras we characterize w continuous markovian semigroups on a von neumann algebra m which are 0 symmetric w r t a faithful normal state 0in m in terms of quadratic forms on the hilbert space h of a standard form m h p j we characterize also symmetric strongly continuous contraction semigroups on a real hilbert space h which leave invariant a closed convex set in h in terms of a contraction property of the associated quadratic forms we apply the results to give criteria of essential selfadjointness for quadratic form sums and to give a characterization of w continuous markovian semigroups on m which commute with the modular automorphism group 0t

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

integrals and potentials of differential 1 forms on the sierpinski gasket abstract we provide a definition of the integral along paths in the sierpinski gasket k for differential smooth 1 forms associated to the standard dirichlet form e on k we show how this tool can be used to study the potential theory on k in particular we prove i a de rham reconstruction of a 1 form from its periods around lacunas in k ii a hodge decomposition of 1 forms with respect to the hilbertian energy norm iii the existence of potentials of smooth 1 forms on a suitable covering space of k we finally show that this framework provides versions of the de rham duality theorem for the fractal k

symmetries of levy processes on compact quantum groups their markov semigroups and potential theory quantum markov semigroups qms i e strongly continuous semigroups of unital completely positive maps on compact quantum groups are studied we show that translation invariant qmss on the universal or reduced c algebra of a compact quantum group are in one to one correspondence with levy processes on its hopf algebra we use the theory of levy processes on involutive bialgebras to characterize symmetry properties of the associated qms it turns out that the qms is self adjoint resp kms symmetric if and only if the generating functional of the levy process is invariant under the antipode resp the unitary antipode furthermore we study levy processes whose marginal states are invariant under the adjoint action finally some related aspects of the potential theory as dirichlet form derivation and spectral triple are investigated we discuss how the above results apply to compact groups group c algebras of discrete groups free orthogonal quantum groups on and twisted suq 2 quantum group

multiple blow up rates in a coupled heat system with mixed type nonlinearities abstract this paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed type nonlinearities at first the critical exponent is obtained and simply described via a characteristic algebraic system introduced by us then as the main results of the paper three blow up rates are established under different dominations of nonlinearities for the one dimensional case and represented in another characteristic algebraic system in particular it is observed that unlike those in previous literature on parabolic models with absorptions two of the multiple blow up rates obtained here do depend on the absorption exponents in the known works the absorptions affect the blow up criteria the blow up time as well as the initial data required for the blow up of solutions all without changing the blow up rates to our knowledge this is the first example of absorption dependent blow up rates exploiting the significant interactions among diffusions inner absorptions and nonlinear boundary fluxes in the coupled system it is also proved that the blow up of solutions in the model occurs on the boundary only

conjugate gradient type methods and preconditioning in this paper we consider various iterative methods for the numerical solution of very large sparse linear systems of equations which arise in the discretization of partial differential equations as the performance of the methods generally depends on the characteristics of the problems to be solved a judicious choice between the methods will require knowledge about the system the aim of this paper is to review the properties of the various iteration methods in order to assist the user in making a deliberate selection

l 2 rigidity in von neumann algebras we introduce the notion of l2 rigidity for von neumann algebras a generalization of property t which can be viewed as an analogue for the vanishing of 1 cohomology into the left regular representation of a group we show that l2 rigidity passes to normalizers and is satisfied by nonamenable ii1 factors which are non prime have property or are weakly rigid as a consequence we obtain that if m is a free product of diffuse von neumann algebras or if m l where is a finitely generated group with 1 2 0 then any nonamenable regular subfactor of m is prime and does not have properties or t in particular this gives a new approach for showing solidity for a free group factor thus recovering a well known recent result of n ozawa

a domain decomposition solver for acoustic scattering by elastic objects in layered media a finite element solution procedure is presented for accurately computing time harmonic acoustic scattering by elastic targets buried in sediment an improved finite element discretization based on trilinear basis functions leading to fourth order phase accuracy is described for sufficiently accurate discretizations 100 million to 1 billion unknowns are required the resulting systems of linear equations are solved iteratively using the gmres method with a domain decomposition preconditioner employing a fast direct solver due to the construction of the discretization and preconditioner iterations can be reduced onto a sparse subspace associated with the interfaces numerical experiments demonstrate capability to evaluate the scattered field with hundreds of wavelengths

a block centered finite difference method for reduced fracture model in karst aquifer system abstract a block centered finite difference scheme is given for the approximation of reduced coupled model in the fractured media aquifer system the fluid flow of aquifer system is governed by darcy s law in both fracture and surrounding porous media the degrees of freedom are given separately on both sides of fracture to capture the discontinuity of pressure and velocity optimal error estimates in discrete l 2 norms are obtained numerical experiments are provided to verify the second order accuracy of presented method it is demonstrated whether the fracture acts as a fast pathway or geological barrier is totally determined by the value of its permeability tensor

roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition in this paper we investigates the blow up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition we obtain the blow up condition and its blow up rate estimate

the biot darcy brinkman model of flow in deformable double porous media homogenization and numerical modelling abstract in this paper we present the two level homogenization of the flow in a deformable double porous structure described at two characteristic scales the higher level porosity associated with the mesoscopic structure is constituted by channels in a matrix made of a microporous material consisting of elastic skeleton and pores saturated by a viscous fluid the macroscopic model is derived by the homogenization of the flow in the heterogeneous structure characterized by two small parameters involved in the two level asymptotic analysis whereby a scaling ansatz is adopted to respect the pore size differences the first level upscaling of the fluid structure interaction problem yields a biot continuum describing the mesoscopic matrix coupled with the stokes flow in the channels the second step of the homogenization leads to a macroscopic model involving three equations for displacements the mesoscopic flow velocity and the micropore pressure due to interactions between the two porosities the macroscopic flow is governed by a darcy brinkman model comprising two equations which are coupled with the overall equilibrium equation respecting the hierarchical structure of the two phase medium expressions of the effective macroscopic parameters of the homogenized double porosity continuum are derived depending on the characteristic responses of the mesoscopic structure some symmetry and reciprocity relationships are shown and issues of boundary conditions are discussed the model has been implemented in the finite element code sfepy which is well suited for computational homogenization a numerical example of solving a nonstationary problem using mixed finite element method is included

a multiscale darcy brinkman model for fluid flow in fractured porous media the aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures the darcy s law is assumed in the porous domain while the stokes brinkman equations are considered in the fractures we address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain and describe the fracture as if it were an interface between porous regions we derive the corresponding interface model governing the fluid flow in the fracture and in the porous media and establish the well posedness of the coupled problem further we introduce a finite element scheme for the approximation of the coupled problem and discuss solution strategies we conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non reduced one

non simultaneous blow up in a parabolic system with three components abstract this paper deals with heat equations coupled via nonlinear boundary flux u 1 u 1 p 11 u 2 p 12 u 2 u 2 p 22 u 3 p 23 u 3 u 3 p 33 u 1 p 31 a necessary and sufficient condition for the existence of only one component blowing up for nondecreasing in time and radially symmetric solutions three kinds of exponent regions are obtained as follows i only one component blows up for every initial data ii the existence of two components blowing up simultaneously while the third one remains bounded also with two kinds of blow up rates 1 2 p 22 1 1 2 p 33 1 and p 23 1 p 33 2 p 33 1 1 2 p 33 1 iii e g u 1 remains bounded and u 2 u 3 blow up simultaneously with blow up rate p 23 1 p 33 2 p 33 1 1 2 p 33 1 for every initial data moreover the eight kinds of simultaneous blow up rates and blow up sets are discussed

a discontinuous finite volume method for a coupled fracture model abstract in this paper we present a discontinuous finite volume method for a hybrid dimensional coupled fracture model on triangular meshes in matrix and fracture the pressure is approximated by the discontinuous and continuous piecewise linear elements and in order to retain local conservation property the velocity is sought in the discontinuous lowest order raviart thomas element space and piecewise linear element space the proposed numerical scheme is only associated with the pressure so we avoid solving an indefinite saddle point problem in addition the discontinuous method eliminates the continuity of approximate functions across the interelement boundary so it has the advantage of high localizability and easy handling of complicated geometries optimal order error estimates for the pressure and velocity are proved and numerical experiments are carried out to confirm our theoretical analysis

blow up phenomena for a system of semilinear heat equations with nonlinear boundary flux abstract this paper deals with the blow up for a system of semilinear r laplace heat equations with nonlinear boundary flux it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time and when blow up does occur upper and lower bounds for the blow up time are obtained

a two level four color sor method a two level four color sor method is proposed for the nine point discretization of the poisson equation on a square instead of examining the jacobi iteration matrix in the space domain we consider an equivalent but much simpler four color iteration matrix in the frequency domain a two level sor method is introduced to increase the convergence rate for the frequency domain iteration matrix at the first level the red and orange points and then the black and green points are treated as groups and a block sor iteration is performed on these two groups at the second level another sor iteration is used to decouple values at the red and orange points and then at the black and green points the conventional red black sor iteration for a five point stencil is shown to be a degenerate case of the general two level four color sor method for the case of the nine point stencil a closed form expression for the optimal relaxation parameters omega b and omega p at the two iteration levels is

multiple absorption related blow up rates to a coupled heat system this paper deals with asymptotic behavior of solutions to a heat system with absorptions and coupling positive multi nonlinearities it is known that although absorption mechanisms may affect such as blow up criteria blow up time and initial data required for blow up solutions they cannot change blow up rates of solutions in general it has been reported in the current literature that blow up rates for scalar equations with absorptions are all absorption independent in a previous paper of the authors four absorption independent simultaneous blow up rates were obtained already for the same problem under weak absorptions the present paper will furthermore prove that if the absorptions are unbalanced in the model i e the absorption is stronger for one component and weaker for another then there are in addition eight possible absorption related blow up rates for the model besides the four absorption independent ones this exposes a significant difference between scalar and coupled nonlinear parabolic equations with absorptions

domain decomposition preconditioners for p and hp finite element approximation of stokes equations abstract domain decomposition preconditioning techniques are developed in the context of hp finite element approximation of the stokes problem two basic types of preconditioner are considered a block diagonal scheme based on decoupling the velocity and pressure components and a scheme based on an indefinite system similar to the original stokes system for each type of scheme theoretical estimates are obtained for the location of the eigenvalues of the preconditioned operators in terms of the polynomial degree the mesh sizes on the coarse and fine grids and the inf sup constant for the method theoretical estimates show that the growth of the bounds is modest as the mesh is refined and the polynomial order is increased the preconditioners are shown to be applicable to various iterative schemes for the stokes systems the theoretical bounds are compared with actual quantities obtained in practical computations for several representative problems

schwarz preconditioners for the spectral element discretization of the steady stokes and navier stokes equations the q n q n 2 spectral element discretization of the stokes equation gives rise to an ill conditioned indefinite symmetric linear system for the velocity and pressure degrees of freedom we propose a domain decomposition method which involves the solution of a low order global and several local problems related to the vertices edges and interiors of the subdomains the original system is reduced to a symmetric equation for the velocity which can be solved with the conjugate gradient method we prove that the condition number of the iteration operator is bounded from above by c 1 log n 3 beta n where c is a positive constant independent of the degree n and the number of subdomains and beta n is the inf sup condition of the pair q n q n 2 we also consider the stationary navier stokes equations in each newton step a non symmetric indefinite problem is solved using a schwarz preconditioner by using an especially designed low order global space and the solution of local problems analogous to those decribed above for the stokes equation we are able to present a complete theory for the method we prove that the number of iterations of the gmres method at each newton step is bounded from above by c 1 log n 3 beta n the constant c does not depend on the number of subdomains or n and it does not deteriorate as the newton iteration proceeds

a fast poisson solver of arbitrary order accuracy in rectangular regions in this paper we propose a direct method for the solution of the poisson equation in rectangular regions it has an arbitrary order accuracy and low cpu requirements which makes it practical for treating large scale problems the method is based on a pseudospectral fourier approximation and a polynomial subtraction technique fast convergence of the fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions these functions have the same discontinuities at the corner points as the sought solution in addition to this they satisfy the laplace equation so that the subtraction procedure does not generate nonperiodic nonhomogeneous terms the solution of a boundary value problem is obtained in a series form in o n log n floating point operations where n2 is the number of grid nodes evaluating the solution at all n2 interior points requires o n2 log n operations

the blow up rate for a strongly coupled system of semilinear heat equations with nonlinear boundary conditions abstract the paper deals with the blow up rate of positive solutions to the system u t u xx u l 11 v l 12 v t v xx u l 21 v l 22 with boundary conditions u x 1 t u p 11 v p 12 1 t and v x 1 t u p 21 v p 22 1 t under some assumptions on the matrices l l ij and p p ij and on the initial data u 0 v 0 the solution u v blows up at finite time t and we prove that max x 0 1 u x t resp max x 0 1 v x t goes to infinity as t t 1 2 resp t t 2 2 where i p id 1 2 t 1 1 t

waveform relaxation with fast direct methods as preconditioner for a restricted class of parabolic pdes one can devise a practical numerical solver with a parallel complexity that is theoretically optimal the method uses a multidimensional fft to decouple the unknowns in the spatial domain into independent scalar odes these are discretized to give recurrence relations in the time dimension solved by parallel cyclic reduction this is the fft cr algorithm we discuss the use of fft cr as a preconditioner to iteratively solve more general parabolic pdes this approach naturally leads to a waveform relaxation scheme waveform relaxation was developed as an iterative method for solving large systems of odes it is the continuous in time analogue of stationary iterative methods for linear algebraic equations using the fft cr solver as a preconditioner preserves most of the potential for concurrency that accounts for the attractiveness of waveform relaxation with simple preconditioners like jacobi or red black gauss seidel while showing an important advantage the convergence rate of the resulting iteration is independent of the mesh size used in the spatial discretization the method can be accelerated by applying an appropriate scaling of the system before preconditioning

is modified psd equivalent to modified sor for two cyclic matrices in this paper we study the convergence analysis of the modified preconditioned simultaneous displacement mpsd method when a is a two cyclic matrix convergence conditions and optimum values of the parameters are determined in case the eigenvalues of the associated jacobi iteration matrix are either all real or all imaginary

a converse to the jensen inequality its matrix extensions and inequalities for minors and eigenvalues we obtain a scalar inequality converse to the jensen inequality we also derive an operator converse to the jensen inequality as special cases we obtain inequalities similar to the kantorovich one as well as some operator generalizations of them using some exterior algebra we prove a generalization of the sylvester determinant theorem we also deduce some determinant analogs of the additive and multiplicative kantorovich inequalities

blow up for a semilinear reaction diffusion system coupled in both equations and boundary conditions we study the blow up behavior for a semilinear reaction diffusion system coupled in both equations and boundary conditions the main purpose is to understand how the reaction terms and the absorption terms affect the blow up properties we obtain a necessary and sufficient condition for blow up derive the upper bound and lower bound for the blow up rate and find the blow up set under certain assumptions

positive solutions for a system of fractional differential equations with coupled integral boundary conditions we investigate the existence of positive solutions for a system of nonlinear riemann liouville fractional differential equations with coupled integral boundary conditions

blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions abstract this paper concerns with blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space we establish the rate estimates for blow up solutions and prove that the blow up set is r n under proper conditions on initial data furthermore for n 1 more complete conclusions about such two topics are given

solution of elliptic pdes by fast poisson solvers using a local relaxation factor abstract using a local discrete fourier analysis one and two step iterative procedures are developed to solve a large class of 2 d and 3 d nonseparable elliptic partial differential equations pdes the one step procedure is a modified d yakanov gunn iterative procedure in which the relaxation factor is grid point dependent the two step procedure is designed to accelerate the one step procedure both are easy to implement and applicable to a variety of boundary conditions they are also computationally efficient as indicated by the results of numerical comparisons with other established methods furthermore these new algorithms possess two important properties which the traditional iterative methods lack i e i the convergence rate is relatively insensitive to grid cell size and aspect ratio and ii the convergence rate can be easily estimated using the coefficient of the pde being solved for a set of constant coefficient model problems it is show theoretically that the two step vs one step computational efficiency ratio ranges between 1 and 2 the higher ratio is realized whenever the need to accelerate the one step procedure is greater i e whenever its convergence rate is lower it is also shown numerically that the two step procedure can substantially outperform the one step procedure in the numerical solution of many pdes with variable coefficients

non simultaneous blow up in coupled heat equations with multi nonlinearities abstract this paper studies coupled heat equations with multi nonlinearities of six nonlinear parameters the critical blow up exponent is established via a complete classification for all the six nonlinear parameters where a precise analysis on the geometry of and the absorption coefficients is given for the balanced interaction situation among the multi nonlinearities the main attention is contributed to non simultaneous phenomena in the model to determine the necessary and sufficient conditions of non simultaneous blow up with suitable initial data as well as the conditions under which any blow up must be non simultaneous finally the model of the paper is characterized by a comparison with the known results for other models

a note about proving non γ under a finite non microstates free fisher information assumption we prove that if x1 xn n 1 are self adjoints in a w probability space with finite non microstates free fisher information then the von neumann algebra w x1 xn they generate doesn t have property especially is not amenable this is an analog of a well known result of voiculescu for microstates free entropy we also prove factoriality under finite non microstates entropy

variations in noncommutative potential theory finite energy states potentials and multipliers in this work we undertake an extension of various aspects of the potential theory of dirichlet forms from locally compact spaces to noncommu tative c algebras with trace in particular we introduce finite energy states potentials and multipliers of dirichlet spaces we prove several results among which the celebrated deny s embedding theorem and the deny s inequality the fact that the carre du champ of bounded potentials are finite energy func tionals and the relative supply of multipliers 1 introduction and description of the results in the present work we develop further the potential theory of dirichlet forms on noncommutative c algebras with trace we introduce and investigate finite energy states potentials and multipliers objects naturally associated to dirichlet spaces and which are meant to encode or reveal the geometric nature of the latter in a companion work the results here obtained will be crucial to construct on c algebras endowed with a dirichlet form the building blocks of a metric differential geometry dirac operators and spectral triples and topological invariants sum mable fredholm modules in k homology in the framework of the noncommutative geometry developed by a connes co classical potential theory studying harmonic functions on euclidean spaces r n finite energy measures and their potentials was based on the properties of kernel x y 1 the so called green function to understood as the integral kernel of the

a two grid decoupled algorithm for fracture models abstract in this paper we consider a two grid decoupled algorithm for two categories of fracture models in the mixed darcy darcy fracture model fluid flow in fractures as well as in the surrounding medium is governed by darcy s law and in the mixed darcy forchheimer darcy fracture model flow in fractures is governed by darcy forchheimer s law while that in surrounding matrix is governed by darcy s law in the proposed two grid method we use a coarse grid approximation to the interface coupling conditions for decoupling mixed problems in fractures and surrounding matrix error estimates show that the two grid decoupled algorithm retains the same order of approximation accuracy as the coupled algorithm for the pressure p on h 1 semi norm and the velocity u on l 2 2 norm numerical experiments are carried out to verify the accuracy and efficiency of the decoupled algorithm in which the computation times are reduced greatly compared to the coupled method

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

a trace theorem for the dirichlet form on the sierpinski gasket in connection with the theory for brownian motion on fractals a corresponding dirichlet form has been defined we consider here the fractal known as the sierpinski gasket and characterize the tra

algebraic two level preconditioners for the schur complement method the solution of elliptic problems is challenging on parallel distributed memory computers since their green s functions are global to address this issue we present a set of preconditioners for the schur complement domain decomposition method they implement a global coupling mechanism through coarse space components similar to the one proposed in bramble pasciak and shatz math comp 47 1986 pp 103 134 the definition of the coarse space components is algebraic they are defined using the mesh partitioning information and simple interpolation operators these preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration the numerical and parallel scalability of those preconditioners are illustrated on two dimensional model examples that have anisotropy and or discontinuity phenomena

an ad hoc sor method abstract an ad hoc method is presented for determining a sequence of successive overrelaxation factors that although having no rigorous foundations at the moment appears to produce a highly competitive iteration scheme for several classes of difference equations a different is chosen for each difference equation

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

a lagrange multiplier method for a stokes biot fluid poroelastic structure interaction model we study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium the free fluid is governed by the stokes equations while the flow in the poroelastic medium is modeled using the biot poroelasticity system equilibrium and kinematic conditions are imposed on the interface a mixed darcy formulation is employed resulting in continuity of flux condition of essential type a lagrange multiplier method is employed to impose weakly this condition a stability and error analysis is performed for the semi discrete continuous in time and the fully discrete formulations a series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters

markov semigroups on c bundles abstract we develop a non commutative theory of dirichlet forms and symmetric markov semigroups acting on the smooth sections of a c algebra bundle we show that the bochner laplacian associated to a suitable connection on a clifford algebra bundle induces a dirichlet form and describe some properties of the associated markov semigroup

an efficient direct parallel spectral element solver for separable elliptic problems an efficient direct parallel elliptic solver based on the spectral element discretization is developed the direct solver is based on a matrix decomposition approach which reduces multi dimensional separable problems to a sequence of one dimensional problems that can be efficiently handled by a static condensation process thanks to the spectral accuracy and the localized nature of a spectral element discretization this elliptic solver is spectrally accurate and can be efficiently parallelized and it can serve as an essential building block for large scale high performance solvers in computational fluid dynamics and computational materials science

derivations as square roots of dirichlet forms abstract the aim of this work is to analyze the structure of a tracially symmetric dirichlet form on a c algebra in terms of a killing weight and a closable derivation taking values in a hilbert space with a bimodule structure it is shown that the generator of the associate markovian semigroup always appears in a natural way as the divergence of a closable derivation applications are shown to the decomposition of dirichlet forms and to the construction of differential calculus on metric spaces

high performance modeling acoustic and elastic waves using the parallel dichotomy algorithm a high performance parallel algorithm is proposed for modeling the propagation of acoustic and elastic waves in inhomogeneous media an initial boundary value problem is replaced by a series of boundary value problems for a constant elliptic operator and different right hand sides via the integral laguerre transform it is proposed to solve difference equations by the conjugate gradient method for acoustic equations and by the gmres k method for modeling elastic waves a preconditioning operator was the laplace operator that is inverted using the variable separation method the novelty of the proposed algorithm is using the dichotomy algorithm 26 which was designed for solving a series of tridiagonal systems of linear equations in the context of the preconditioning operator inversion via considering analytical solutions it is shown that modeling wave processes for long instants of time requires high resolution meshes the proposed parallel fine mesh algorithm enabled to solve real application seismic problems in acceptable time and with high accuracy by solving model problems it is demonstrated that the considered parallel algorithm possesses high performance and efficiency over a wide range of the number of processors from 2 to 8192

an existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions the existence and uniqueness of positive solutions for a second order nonlinear system with coupled integral boundary value conditions are investigated the main tools used in the proofs are a priori estimate method and a maximal principle

iterative methods for cyclically reduced non self adjoint linear systems we study iterative methods for solving linear systems of the type arising from two cyclic discretizations of non self adjoint two dimensional ellip tic partial differential equations a prototype is the convection diffusion equa tion the methods consist of applying one step of cyclic reduction resulting in a reduced system of half the order of the original discrete problem com bined with a reordering and a block iterative technique for solving the reduced system for constant coefficient problems we present analytic bounds on the spectral radii of the iteration matrices in terms of cell reynolds numbers that show the methods to be rapidly convergent in addition we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the unreduced system

blow up estimates of the positive solution of a parabolic system this paper establishes the blow up estimates for the systems ut u 0 vt v 0 in br 0 t br rn with the nonlinear boundary conditions u um1vn1 and v um2vn2 on sr 0 t where 0 m1 1 m2 and 0 n2 1 n1 we prove that c t t 2 max u x t c t t 2 and c t t 2 max v x t c t t 2 under some monotonicity assumptions on the initial values where n1 n2 1 m2 m1 1 and n1m2 1 m1 1 n2

on the product of convex combinations and the kantorovich inequality abstract let i and i i 1 2 n be real constants and i i 1 2 n nonnegative variables satisfying i 1 it is shown that the n 1 dimensional extremum problem of finding the global maximum and minimum of the product i i i i can be reduced to a one dimensional problem defined on the boundary of the convex polygon spanned by the ordered pairs i i for a given set of positive i s 1 2 n 0 the kantorovich inequality states that max i i i i 1 n 2 4 1 n and min i i i i 1 if i 1 i i 1 2 n it is shown that the above new technique can be used to identify exactly how far the i s can vary from the kantorovich values i 1 i without invalidating the kantorovich bounds let max sup x x 1 x ax x bx and min inf x x 1 x ax x bx where x is an n dimensional complex vector and a and b are two n n positive definite hermitian matrices for the special case in which b a 1 the kantorovich inequality implies that max min a 1 2 4 a where a is the spectral condition number of a it is shown that in general max min max a 1 2 4 a b 1 2 4 b the conditions under which equality occurs are also established

impact of eliminating fracture intersection nodes in multiphase compositional flow simulation algebraic elimination of nodes at discrete fracture intersections via the star delta technique has proven to be a valuable tool for making multiphase numerical simulations more tractable and efficient this study examines the assumptions of the star delta technique and exposes its effects in a 3 d multiphase context for advective and dispersive diffusive fluxes key issues of relative permeability saturation capillary pressure kr s pc and capillary barriers at fracture fracture intersections are discussed this study uses a multiphase compositional finite difference numerical model in discrete fracture network dfn and discrete fracture matrix dfm modes it verifies that the numerical model replicates analytical solutions and performs adequately in convergence exercises conservative and decaying tracer one and two phase flow dfm and dfn domains the study culminates in simulations of a two phase laboratory experiment in which a fluid invades a simple fracture intersection the experiment and simulations evoke different invading fluid flow paths by varying fracture apertures as oil invades water filled fractures and as water invades air filled fractures results indicate that the node elimination technique as implemented in numerical model correctly reproduces the long term flow path of the invading fluid but that short term temporal effects of the capillary traps and barriers arising from the intersection node are lost

a modified sor method for the poisson equation in unsteady free surface flow calculations convergence difficulties that sometimes occur if the successive overrelaxation sor method is applied to the poisson equation on a region with irregular free boundaries are analyzed it is shown that these difficulties are related to the treatment of the free boundaries and caused by the appearance of complex eigenvalues in the system of discrete equations when standard centered differences are used after a modification of this system of equations such that the complex eigenvalues become small a modified sor method is presented where two relaxation factors are used alternately the method leads to fast convergence without requiring specific information about the complex eigenvalues

finite element approximation of elliptic problems with dirac measure terms in weighted spaces applications to one and three dimensional coupled problems in this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving dirac measures using weighted sobolev spaces and weighted discrete norms our approach handles both the cases where the measure is simply a right hand side and where it represents an additional term i e solution dependent in the formulation of the problem the main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains where the coupling terms are seen as dirac measures concentrated on the fractures we first establish a decomposition lemma which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces then we consider the stability of the galerkin approximation with finite elements in weighted norms with uniform and graded meshes we introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf sup conditions in weighted norms then we focus on the convergence of the finite element method due to the lack of regularity the convergence rates are suboptimal for uniform meshes we show that using graded meshes optimal rates are recovered our theoretical results are supported by several numerical experiments finally we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three dimensional media with one dimensional fractures that are found in the analysis of microvascular flows

analysis of the coupled navier stokes biot problem abstract we analyze a weak formulation of the coupled problem defining the interaction between a free fluid and a poroelastic structure the problem is governed by the time dependent incompressible navier stokes equations and the biot equations under a small data assumption existence and uniqueness results are proved and a priori estimates are provided

performance of certain iterative methods in solving implicit difference schemes for 2 d navier stokes equations the performance of three iterative methods local sor preconditioned generalized conjugate gradient methods such as orthomin 1 and the gcg ls method with preconditioning matrices derived by incomplete modified and unmodified pointwise factorizations of the matrix corresponding to implicit finite difference schemes for the 2 d navier stokes equations is compared the relative merits of these methods in terms of cpu time average number of iterations with respect to the reynolds number and discretization parameters are emphasized

a darcy brinkman model of fractures in porous media abstract for a fully coupled darcy stokes system describing the exchange of fluid and stress balance across the interface between a saturated porous medium and an open very narrow channel the limiting problem is characterized as the width of the channel converges to zero it is proven that the limit problem is a fully coupled system of darcy flow in the porous medium with brinkman flow in tangential coordinates of the lower dimensional interface

blow up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition this paper is devoted to a nonlinear parabolic equation subject to nonlocal boundary condition and with nonlocal reaction source some criteria on this problem which determine whether the solutions blow up in a finite time or exist globally are given moreover the blow up rate estimates are also obtained

on local relaxation methods and their application to convection diffusion equations abstract this paper discusses local relaxation lr methods which can be regarded as generalizations of the successive overrelaxation sor method the difference is that within an lr method the relaxation factor is allowed to vary from equation to equation a number of existing methods are found to be in fact special lr methods moreover based on sor theory a new lr method is developed the performance of lr methods is illustrated by applying them to central difference approximations of convection diffusion equations it is found that equations with small diffusion coefficients can be handled without difficulty for equations with strongly varying coefficients and for nonlinear equations a properly selected lr method can be significantly more efficient than the optimum sor method as a special example a 16 16 driven cavity problem for a reynolds number of 10 6 can be solved in just a few seconds on a modern computer

global blow up for a localized nonlinear parabolic equation with a nonlocal boundary condition abstract this paper deals with the blow up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition under certain conditions the blowup criteria is established furthermore when f u u p 0 p 1 the global blowup behavior is shown and the blowup rate estimates are also obtained

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem