information loss in coarse graining of stochastic particle dynamics recently a new class of approximating coarse grained stochastic processes and associated monte carlo algorithms were derived directly from microscopic stochastic lattice models for the adsorption desorption and diffusion of interacting particles 12 13 15 the resulting hierarchy of stochastic processes is ordered by the level of coarsening in the space time dimensions and describes mesoscopic scales while retaining a significant amount of microscopic detail on intermolecular forces and particle fluctuations here we rigorously compute in terms of specific relative entropy the information loss between non equilibrium exact and approximating coarse grained adsorption desorption lattice dynamics our result is an error estimate analogous to rigorous error estimates for finite element finite difference approximations of partial differential equations we prove this error to be small as long as the level of coarsening is small compared to the range of interaction of the microscopic model this result gives a first mathematical reasoning for the parameter regimes for which approximating coarse grained monte carlo algorithms are expected to give errors within a given tolerance

estimates for p harmonic functions vanishing on a flat we study p harmonic functions in a domain crn near an m dimensional plane an m flat m where 0 m n 1 in particular let u be a positive p harmonic function with n p vanishing on a porti

radial limits of m subharmonic functions m subharmonic functions are defined in the unit ball of cn their basic properties are developed leading to the following generalization of a theorem of littlewood an m subharmonic function such that its restrictions to spheres centered at the origin are bounded in l1 must have radial limits almost everywhere on the unit sphere the purpose of the present paper is to define and develop the basic properties of the class of m subharmonic functions in the unit ball of cn see definition 1 15 this is the version of subharmonic associated with m harmonic as in ru the paper is divided into four sections 1 consists of preliminary definitions and a few basic results concluding with a characterization of those m subharmonic functions which possess least m harmonic majorants theorems 1 22 and 1 23 2 proves the riesz decomposition theorem for m subharmonic functions with a least m harmonic majorant theorem 2 16 3 contains our main theorem an m subharmonic function satisfying the appropriate growth condition has radial limits almost everywhere theorem 3 1 in the case n 1 this was proved by littlewood lw 4 shows via a very simple example that theorem 3 1 does not extend to admissible limits nor even to nontangential limits nor to any sort of approach strictly more general than radial note however that an m subharmonic function satisfying stronger growth conditions than those in theorem 3 1 must have admissible limits a e see cs this paper is a revised version of the author s thesis written at the university of wisconsin under the direction of professor walter rudin it is a pleasure to record here my gratitude for many stimulating discussions on this and various other topics 1 preliminaries notations are for the most part as in ru throughout this paper n will be a fixed positive integer usually n 1 our results are all standard when n 1 including the case n 1 would occasionally necessitate dividing our notation if not our argument into two cases for z w e cn set z w ej lzjwj and z z z 1 2 let b z e cn lzl 1 and s t e c ldl 1 for z e cn we will sometimes write z zl z where z1 e c z e c 1 for z a e b a o define a pa z 1 a qa z 4 a z 1 zs a received by the editors october 18 1984 1980 mathematicns subject classification primary 32m15 secondary 31b25 c 1985 american mathematical society 0002 9947 85 1 00 25 per page 501 this content downloaded from 207 46 13 122 on thu 19 may 2016 04 17 58 utc all use subject to http about jstor org terms

the scaled flexibility matrix method for the efficient solution of boundary value problems in 2d and 3d layered elastic media we present a method that extends the flexibility matrix method for multilayer elasticity problems to include problems with very thin layers this method is particularly important for solving problems in which one or a number of very thin layers are juxtaposed with very thick layers the standard flexibility matrix method suffers from round off errors and poor scaling of the flexibility equations which occur when one of the layers in the multilayered medium becomes much smaller than the others the method proposed in this paper makes use of power series expansions of the various components of the flexibility matrix in order to arrive at a system of equations that is appropriately scaled the effectiveness of the scaled flexibility matrix method is demonstrated on a number of test problems

self similar solutions for a fractional thin film equation governing hydraulic fractures in this paper self similar solutions for a fractional thin film equation governing hydraulic fractures are constructed one of the boundary conditions which accounts for the energy required to break the rock involves the toughness coefficient k geq 0 mathematically this condition plays the same role as the contact angle condition in the thin film equation we consider two situations the zero toughness k 0 and the finite toughness k in 0 infty cases in the first case we prove the existence of self similar solutions with constant mass in the second case we prove that for all k 0 there exists an injection rate for the fluid such that self similar solutions exist

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

a semicoarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients the purpose of this paper is to present a semicoarsening multigrid algorithm for solving the finite difference discretization of symmetric and nonsymmetric two and three dimensional elliptic partial differential equations with highly discontinuous and anisotropic coefficients the discrete equations are assumed to be defined on a logically rectangular grid obtained possibly through grid generation for a problem defined on an irregular domain the basic algorithm is described along with some modifications which are designed to improve its efficiency and robustness for certain types of problem cases fortran codes that implement the two and three dimensional semicoarsening multigrid algorithms are described briefly and numerical results are presented

information loss in coarse graining of polymer configurations via contact matrices contact matrices provide a coarse grained description of the configuration of a linear chain polymer or random walk on n ij 1 when the distance between the positions of the ith and jth steps are less than or equal to some distance a and ij 0 otherwise we consider models in which polymers of length n have weights corresponding to simple and self avoiding random walks srw and saw with a the minimal permissible distance we prove that to leading order in n the number of matrices equals the number of walks for srw but not for saw the coarse grained shannon entropies for srw agree with the fine grained ones for n 2 but differs for n 3

the black box multigrid numerical homogenization algorithm in mathematical models of flow through porous media the coefficients typically exhibit severe variations in two or more significantly different length scales consequently the numerical treatment of these problems relies on ahomogenizationorupscalingprocedure to define an approximate coarse scale problem that adequately captures the influence of the fine scale structure inherent in such a procedure is a compromise between its computational cost and the accuracy of the resulting coarse scale solution although techniques that balance the conflicting demands of accuracy and efficiency exist for a few specific classes of fine scale structure e g fine scale periodic this is not the case in general in this paper we propose a new efficient numerical approach for thehomogenizationof the permeability in models of single phase saturated flow our approach is motivated by the observation that multiple length scales are captured automatically by robust multilevel iterative solvers such as dendy sblack box multigrid in particular the operator induced variational coarsening in black box multigrid produces coarse grid operators that capture the essential coarse scale influence of the medium s fine scale structure we derive an explicit local cell based approximate expression for the symmetric 2 2 homogenized permeability tensor that is defined implicitly by the black box coarse grid operator the effectiveness of this black box multigrid numerical homogenization method is demonstrated through numerical examples

a large deviation result for the range of random walk and for the wiener sausage let s n be a random walk on d and let r n be the number of different points among 0 s1 s n 1 we prove here that if d 2 then x lim n 1 n logp r n nx exists for x 0 and establish some convexity and monotonicity properties of x the one dimensional case will be treated in a separate paper

sobolev s inequalities and vanishing integrability for riesz potentials of functions in the generalized lebesgue space lp logl q abstract our aim in this paper is to deal with sobolev s inequalities for riesz potentials of functions belonging to l p log l q to do so we study the boundedness of hardy littlewood maximal functions and apply the hedberg s trick as an application we treat vanishing integrability for riesz potentials

self similar solutions for elastohydrodynamic cavity flow the enlargement of a lens shaped cavity lying in a plane of cleavage between two elastic half spaces and filling with viscous fluid from a source on the axis of symmetry is considered the internal flow is modelled by lubrication theory which gives a nonlinear partial differential equation connecting the pressure to the cavity shape and the same two quantities are also related by the singular integral equation of linear elasticity if the total volume of fluid q t in the cavity at time t is proportional either to t or to exp t the resulting boundary value problem can be reduced to a self similar form in which time does not appear explicitly the solution in non dimensional terms depends on a single parameter which may be interpreted as the stress intensity factor k at the tip calculations have been made for the two dimensional version of the problem for a range of values of and for a range of stress intensities the numerical method is to expand the cavity height in a chebyshev series the coefficients being found by a nonlinear optimization technique to yield a least squares fit to the reynolds equation these lead to expressions for the rate of cavity growth and other quantities of physical interest

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

a new multigrid approach to convection problems abstract a new multigrid method for convection problems is presented it is designed to overcome the problem of alignment the flow being aligned with the grid the technique employs semi coarsening in several directions simultaneously and gives rise to multiple coarser grids with the same total number of points per grid level but with different sizes in each co ordinate direction the amount of work per multigrid cycle is still o n as an example the method is applied to the nonlinear upwind differenced euler equations of gas dynamics in two dimensions convergence rates are estimated by two level fourier analysis for the linearised equations numerical experiments on the nonlinear equations confirm these estimates

growth properties near the origin for generalized riesz potentials abstract according to riesz decomposition theorem superharmonic functions on the punctured unit ball are represented as the sum of generalized potentials and harmonic functions in this paper we study growth properties near the origin of spherical means for generalized riesz potentials of functions belonging to central variable morrey spaces we also deal with monotone sobolev functions

growth properties for riesz potentials of functions in weighted variable lp spaces abstract we study growth properties of spherical means for riesz potentials of functions in weighted lebesgue spaces of variable exponent we also deal with green potentials and monotone sobolev functions

an implicit level set method for modeling hydraulically driven fractures we describe a novel implicit level set algorithm to locate the free boundary for a propagating hydraulic fracture a number of characteristics of the governing equations for hydraulic fractures and their coupling present considerable challenges for numerical modeling namely the degenerate lubrication equation the hypersingular elastic integral equation the indeterminate form of the velocity of the unknown fracture front which precludes the implementation of established front evolution strategies that require an explicit velocity field and the computationally prohibitive cost of resolving all the length scales an implicit algorithm is also necessary for the efficient solution of the stiff evolution equations that involve fully populated matrices associated with the coupled non local elasticity and degenerate lubrication equations the implicit level set algorithm that we propose exploits the local tip asymptotic behavior applicable at the computational length scale in order to locate the free boundary local inversion of this tip asymptotic relation yields the boundary values for the eikonal equation whose solution gives the fracture front location as well as the front velocity field the efficacy of the algorithm is tested by comparing the level set solution to analytic solutions for hydraulic fractures propagating in a number of distinct regimes the level set algorithm is shown to resolve the free boundary problem with first order accuracy further it captures the field variables such as the fracture width with the first order accuracy that is consistent with the piecewise constant discretization that is used

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

harnack s inequality for p harmonic functions with unbounded exponent p abstract we study properties of the function u lim u where u is the solution of the min p laplacian dirichlet problem with bounded sobolev boundary function here p n is a variable exponent such that 1 p is lipschitz continuous we derive bloch type estimates and using them we prove harnack s inequality in cases of unbounded but finite exponent

enrichment strategies and convergence properties of the xfem for hydraulic fracture problems abstract in two recent papers gordeliy and peirce 2013 investigating the use of the extended finite element method xfem for modeling hydraulic fractures hf two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful xfem hf algorithms in this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies in addition we derive a novel set of crack tip enrichment functions that enable the xfem to model hf with the full range of power law r behavior of the displacement field and the corresponding r 1 singularity in the stress field for 1 2 1 this novel crack tip enrichment enables the xfem to achieve the optimal convergence rate which is not achieved by existing enrichment functions used for this range of power law the two xfem boundary value problem classes are as follows i a neumann to dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the xfem is used to solve for the corresponding crack width p w and ii a mixed hybrid formulation of the xfem that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it p w the two enrichment schemes considered are i the xfem t scheme with full singular crack tip enrichment and ii a simpler more efficient xfem s scheme in which the singular tip behavior is only imposed in a weak sense if enrichment is applied to all the nodes of tip enriched elements then the resulting xfem stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions which is a situation that also holds for the classic set of square root enrichment functions for the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix numerical experiments indicate that the xfem t scheme with the new tip enrichment achieves the optimal o h 2 convergence rate we expect of the underlying piece wise linear fem discretization which is superior to the enrichment functions currently available in the literature for 1 2 1 the xfem s scheme with only signum enrichment to represent the crack geometry achieves an o h convergence rate it is also demonstrated that the standard p w formulation based on the variational principle of minimum potential energy is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce while the mixed hybrid p w formulation based on the hellinger reissner variational principle does not have this disadvantage

capacity for potentials of functions in musielak orlicz spaces abstract we define a capacity for potentials of functions in musielak orlicz spaces basic properties of such capacity are studied we also estimate the capacity of balls and give some applications of the estimates

modeling multi scale processes in hydraulic fracture propagation using the implicit level set algorithm abstract in this paper we describe an implicit level set algorithm ilsa peirce and detournay 2008 suitable for modeling multi scale behavior in planar hydraulic fractures propagating in three dimensional elastic media this multi scale behavior is typically encountered when multiple physical processes compete to determine the location of the fracture free boundary instead of having to match the mesh size to the finest active length scale or having to re mesh as the dominant length scales change in space and time the novel ilsa scheme is able to represent the required multi scale behavior on a relatively coarse rectangular mesh this is achieved by using the local front velocity to construct for each point of a set of control points a mapping that adaptively identifies the dominant length scale at which the appropriate multi scale universal asymptotic solution needs to be sampled finer scale behavior is captured in a weak sense by integrating the universal asymptotic solution for the fracture width over partially filled tip elements and using these integrals to set the average values of the widths in all tip elements the ilsa solution shows good agreement with a multi scale reference solution comprising a radial solution that transitions from viscosity to toughness dominated propagation regimes the ilsa scheme is also used to model blade like hydraulic fractures that break through stress barriers located symmetrically with respect to the injection point for the zero toughness case the ilsa solution shows close agreement to experimental results the multi scale ilsa scheme is also used to provide results when the material toughness k i c is non zero in this case different parts of the fracture free boundary can be propagating in different regimes it is hoped that the multi scale ilsa solutions presented here will form a set of reference results that can be used to benchmark simulators that use a propagation criterion based on only one dissipative process either toughness or viscosity the multi scale ilsa solutions at larger times for which plane strain conditions develop in vertical cross sections are compared with and show close agreement to plane strain exact solutions for height growth and the fracture width in vertical cross sections this comparison provides some measure of the accuracy of the multi scale ilsa scheme the multi scale ilsa solutions are also used to identify the regimes of applicability of pseudo 3d p3d approximate solutions these ilsa solutions can also be used to design improved p3d models

polymers as self avoiding walks a brief overview is presented of the relation of the properties of real polymers to the problem of self avoiding random walks the self consistent field method is discussed wherein the non markovian continuous self avoiding polymer is replaced by a self consistent markovian approximation an outline is presented of the method of solution of the resultant nonlinear integrodifferential equations a description is also presented of the scaling theories which provide a means for deducing some exponents in the asymptotic dependence of walk properties on the length of the walk in the limit of infinite length walks

universal fluctuations in the support of the random walk a random walk starts from the origin of ad dimensional lattice the occupation numbern x t equals unity if aftert steps site x has been visited by the walk and zero otherwise we study translationally invariant sumsm t of observables defined locally on the field of occupation numbers examples are the numbers t of visited sites the areae t of the appropriately defined surface of the set of visited sites and in dimension d 3 the euler index of this surface ind 3 theaverages m t all increase linearly witht ast we show that in d 3 to leading order in an asymptotic expansion int thedeviations from average m t m t m t are up to a normalization allidentical to a single universal random variable this result resembles an earlier one in dimensiond 2 we show that this universality breaks down ford 3

on the continuity of weakly monotone functions abstract a criterion for the continuity of weakly monotone functions is proposed in terms of the decreasing rearrangement of their gradient its use in the proof of the continuity of weakly monotone functions with gradient in rearrangement invariant spaces is also demonstrated in particular weakly monotone functions whose gradient belongs to an orlicz space or to a lorentz space are discussed and results available in the literature in this regard are extended and refined

growth properties of musielak orlicz integral means for riesz potentials abstract in this paper we are concerned with growth properties of integral means for riesz potentials of variable order of functions in musielak orlicz spaces l r n where is a function of the form x t t p x x t here and p satisfy the log holder condition we also discuss the size of the exceptional sets using a capacity introduced in maeda et al 2011

statistical properties of the set of sites visited by the two dimensional random walk we study the support i e the set of visited sites of a t step random walk on a two dimensional square lattice in the large t limit a broad class of global properties m t of the support is considered including for example the number s t of its sites the length of its boundary the number of islands of unvisited sites that it encloses the number of such islands of given shape size and orientation and the number of occurrences in space of specific local patterns of visited and unvisited sites on a finite lattice we determine the scaling functions that describe the averages on appropriate lattice size dependent time scales on an infinite lattice we first observe that the all increase with t as where k is an m dependent positive integer we then consider the class of random processes constituted by the fluctuations around average we show that to leading order as t gets large these fluctuations are all proportional to a single universal random process normalized to for the probability law of tends to that of varadhan s renormalized local time of self intersections an implication is that in the long time limit all are proportional to

a dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems we present a novel multigrid mg procedure for the efficient solution of the large non symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi layered elastic medium the governing equations involve a highly non linear coupled system of integro partial differential equations along with the fracture front free boundary problem the conditioning of the algebraic equations typically degrades as o n3 a number of characteristics of this problem present significant new challenges for designing an effective mg strategy large changes in the coefficients of the pde are dealt with by taking the appropriate harmonic averages of the discrete coefficients coarse level green s functions for multiple elastic layers are constructed using a single dual mesh and superposition coarse grids that are sub sets of the finest grid are used to treat mixed variable problems associated with pinch points localized approximations to the jacobian at each mg level are used to devise efficient gauss seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements the performance of the mg preconditioner is demonstrated in a number of numerical experiments copyright 2005 john wiley sons ltd

weighted sobolev theorem in lebesgue spaces with variable exponent abstract for the riesz potential operator i there are proved weighted estimates i f l q w q p c f l p w r n 1 q x 1 p x n within the framework of weighted lebesgue spaces l p w with variable exponent in case is a bounded domain the order x is allowed to be variable as well the weight functions are radial type functions fixed to a finite point and or to infinity and have a typical feature of muckenhoupt wheeden weights they may oscillate between two power functions conditions on weights are given in terms of their boyd type indices an analogue of such a weighted estimate is also obtained for spherical potential operators on the unit sphere s n r n

radial limits of quasiregular local homeomorphisms we study the boundary behavior of spatial quasiregular local homeomorphisms defined on the unit ball we show that such maps always have radial limits at some boundary points this result is new also in the case of bounded maps the result fails for unbounded planar maps

campanato morrey spaces for the double phase functionals with variable exponents abstract our aim in this paper is to show that the riesz potential operator i of variable order embeds from variable exponent morrey spaces l p g to campanato morrey spaces in the case x p x x our result extends the recent work of rafeiro and samko 2019 and the authors mizuta et al 2020 we show that i embeds from morrey spaces l g of the double phase functionals x t t p x b x t q x to campanato morrey spaces where p and q satisfy log holder conditions p x q x and b is nonnegative bounded and holder continuous of order 0 1 we also discuss the continuity of riesz potentials i f of functions in l g and show that i embeds from l g to vanishing campanato morrey spaces

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

on the rate of growth of the means mp of holomorphic and pluriharmonic functions on the ball a if u e bp and v is a harmonic conjugate of then b theorem a was proved by using a result due to duren et al 12 theorem 51 concerning fractional derivatives and fractional integrals of functions in bp iff z cpzo a k zk is holomorphic in d and 0 then the fractional derivative ft4 and fractional integral fib of order are defined respectively by and

3d corrected xfem approach and extension to finite deformation theory in this paper the modified or corrected extended finite element method originally presented in fries int j numer meth engng 2008 75 503 532 for the 2d case is extended to 3d including different remedies for the problem that the crack front enrichment functions are linearly dependent in the blending elements in the context of this extension we address a number of computational issues of the 3d xfem in particular possible quadrature rules for elements with discontinuities also the influence of finite deformation theory for crack simulations in comparison to linear elastic fracture mechanics is investigated a number of numerical examples demonstrate the behavior of the presented possibilities copyright 2010 john wiley sons ltd

a hermite cubic collocation scheme for plane strain hydraulic fractures we describe a novel cubic hermite collocation scheme for the solution of the coupled integro partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain special blended cubic hermite power law basis functions with arbitrary index 0 1 are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture the implementation of blended infinite elements to model semi infinite crack problems is also described explicit formulae for the integrated kernels associated with the cubic hermite and blended basis functions are provided the cubic hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions the convergence rate of the cubic hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the o h4 error due to the hermite cubic interpolation the errors due to these two approximations need to be matched in order to achieve optimal convergence backward euler time stepping yields a robust algorithm that along with geometric increments in the time step can be used to explore the transition between propagation regimes over many orders of magnitude in time

boundedness of fractional maximal operators for double phase functionals with variable exponents abstract in this paper we study the boundedness of variable fractional maximal operators m for double phase functionals p q x t t p x b x t q x where is a measurable function on r n satisfying 0 inf x r n x sup x r n x n p q are variable exponents satisfying log holder condition and b is a non negative bounded function which is holder continuous of order 0 1 we also discuss sobolev s inequality for variable riesz potentials i f of double phase functionals with variable exponents

a localized mixed hybrid method for imposing interfacial constraints in the extended finite element method xfem the paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method an enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily shaped material interfaces using an unfitted background mesh the use of lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations in a defined region in the vicinity of the interface the original problem is re stated in its auxiliary formulation the availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form the contribution discusses the weak imposition of dirichlet and neumann type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation the properties of the proposed approach applied to two dimensional linear scalar and vector valued elliptic problems are investigated by studying the convergence behavior

statistical properties of contact maps a contact map is a simple representation of the structure of proteins and other chain like macromolecules this representation is quite amenable to numerical studies of folding we show that the number of contact maps corresponding to the possible configurations of a polypeptide chain of n amino acids represented by n 1 step self avoiding walks on a lattice grows exponentially with n for all dimensions d 1 we carry out exact enumerations in d 2 on the square and triangular lattices for walks of up to 20 steps and investigate various statistical properties of contact maps corresponding to such walks we also study the exact statistics of contact maps generated by walks on a ladder

localized jacobian ilu preconditioners for hydraulic fractures we discuss the properties of a class of sparse localized approximations to the jacobian operator that arises in modelling the evolution of a hydraulically driven fracture in a multi layered elastic medium the governing equations involve a highly non linear coupled system of integro partial differential equations along with the fracture front free boundary problem we demonstrate that an incomplete lu factorization of these localized jacobians yields an efficient preconditioner for the fully populated stiff non symmetric system of algebraic equations that need to be solved multiple times for every growth increment of the fracture the performance characteristics of this class of preconditioners is explored via spectral analysis and numerical experiment copyright 2005 john wiley sons ltd

an implicit algorithm for the propagation of a hydraulic fracture with a fluid lag abstract we present an implicit moving mesh algorithm for the study of the propagation of plane strain hydraulic fracture in an impermeable medium in particular the fluid front is allowed to lag behind the fracture tip the solution expressed in the proper scaling evolves for a given value of a dimensionless toughness from a zero stress zero time self similar solution characterized by a finite lag to a large stress large time self similar solution with zero lag a numerical solution for the transient problem is presented using two distinct meshes stretching at different velocities for the lag and fluid filled part of the crack the lubrication and elasticity equations are solved in a coupled manner using an implicit scheme the lag size also coupled to the preceding equations is obtained via the fracture propagation condition the numerical results are discussed and compared with the known zero lag solution for large stress large time