the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

a conforming finite element method for overlapping and nonmatching grids in this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity both overlapping and nonoverlapping cases are considered we prove that the new method admits an optimal convergence rate the error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains our results are valid for multiple subdomains and any spatial dimensions

a multigrid method for the mortar type morley element approximation of a plate bending problem in this paper a multigrid algorithm is presented for the mortar element method for the morley nonconforming plate element based on the multigrid theory developed by bramble pasciak and xu in math comp 56 1991 pp 1 34 we prove that the w cycle multigrid method is optimal i e the convergence rate is independent of the mesh size and mesh level moreover a variable v cycle multigrid preconditioner is constructed which results in a preconditioned system with a uniformly bounded condition number

a new cement to glue nonconforming grids with robin interface conditions the finite element case robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in the convergence non conforming grids make the grid generation much easier and faster since it is then a parallel task the goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of robin interface conditions with non matching grids we consider both a symmetric definite positive operator and the convection diffusion equation discretized by finite volume schemes numerical results are shown

on conservation at grid interfaces this paper considers the solution of hyperbolic systems of conservation laws on discontinuous grids in particular we consider what happens to conservation at grid interfaces a procedure is presented to derive conservative difference approximations at the grid interfaces for two dimensional grids which overlap in an arbitrary configuration the same procedures are applied to compute interface formulas for grids which are refined in space and or time and for continuous grids where a switch in the scheme causes the discontinuity

a feti preconditioner for two dimensional edge element approximations of maxwell s equations on nonmatching grids a class of feti methods for the mortar approximation of a vector field problem in two dimensions is proposed edge element discretizations of lowest degree are considered the method proposed can be employed with geometrically conforming and nonconforming partitions our numerical results show that its condition number increases only with the number of unknowns in each subdomain and is independent of the number of subdomains and the size of the problem

interior regularity estimates for elliptic systems of difference equations we prove interior regularity estimates for a large class of difference approximations for elliptic systems of partial differential equations these estimates are analogous to those for the systems of differential equations from these regularity estimates we obtain estimates on the convergence of the solutions of the difference equations a theory of pseudo difference operators with order is used to prove the regularity results we also comment on factors to be considered in choosing a difference scheme for computations

the mixed mortar finite element method for the incompressible stokes problem convergence analysis our purpose is to prove that the mortar element method using local compatible mixed finite elements provides optimal convergence results when applied to the incompressible stokes equations

a feti dp formulation of three dimensional elasticity problems with mortar discretization in this paper a feti dp formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered to resolve the nonconformity of the finite elements a mortar matching condition is imposed on the subdomain interfaces faces a feti dp algorithm is then built by enforcing the mortar matching condition in dual and primal ways in order to make the feti dp algorithm scalable a set of primal constraints which include average and momentum constraints over interfaces are selected from the mortar matching condition a condition number bound c 1 log h h 2 is then proved for the feti dp formulation for the elasticity problems with discontinuous material parameters only some faces need to be chosen as primal faces on which the average and momentum constraints are imposed numerical results are included

nonoverlapping domain decomposition methods with a simple coarse space for elliptic problems we propose a substructuring preconditioner for solving three dimensional elliptic equations with strongly discontinuous coefficients the new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by bramble pasiack and schatz 1989 but with much simpler coarse solvers though the condition number of the preconditioned system may not have a good bound we are able to show that the convergence rate of the pcg method with such substructuring preconditioner is nearly optimal and also robust with respect to the possibly large jumps of the coefficient in the elliptic equation

on the overlapping grid method for elliptic boundary value problems composite overlapping grid methods were studied by volkov proceedings of the steklov institute of mathematics 96 1968 pp 145 185 and starius numer math 28 1977 pp 243 258 more than 20 years ago more recently several authors have used such methods for a variety of problems not necessarily elliptic bvps this work is motivated by the recent work of chesshire and henshaw j comp phys 90 1990 pp 1 64 who developed general software and have carried out extensive computational studies this report is limited to the case where the finite difference operators are of positive type however within that class of problems we present a general theory

the mortar element method with overlapping subdomains in this paper we discuss an extension of the mortar element method to overlapping subdomains and nonmatching grids in the overlapped zones we discuss in particular the case where more than two subdomains overlap the method is described and analyzed as are some preconditioned iterative methods for solving the linear systems arising from this discretization

conservative load transfer along curved fluid solid interface with non matching meshes we investigate the effect of curvature on the accuracy of schemes used to transfer loads along the interface in coupled fluid solid simulations involving non matching meshes we analyze two types of load transfer schemes for the coupled system a point to element projection schemes and b common refinement schemes the accuracy of these schemes over the curved interface is assessed with the aid of static and transient problems we show that the point to element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution the common refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non matching meshes we show theoretically that the accurate transfer preserves the stability of the coupled system while maintaining the energy conservation over a reference interface finally we introduce simple analytical error functions which correlate well with the numerical errors of the load transfer schemes

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

substructuring preconditioners for the q_1 mortar element method the q 1 mortar element method is a non conforming finite element method with q 1 elements based on domain decomposition for the laplace equation it yields an ill conditioned linear system for solving the linear system the so called preconditioned conjugate gradient method in a subspace is used preconditioners are proposed and estimates on condition numbers and arithmetical complexity are given finally numerical experiments are presented

bv estimates for mortar methods in linear elasticity this paper is concerned with the convergence of mortar methods applied to linear elasticity we prove that the conventional mesh dependent norms used in the analysis of mortar methods are bounded below by the bv norm when combined with standard results this bound establishes a decomposition independent and mesh independent proof of the convergence of mortar methods in linear elasticity

hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods a weak continuity condition at the interfaces is enforced by means of lagrange multipliers the two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical two level splitting the first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the lagrange multiplier it is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted l2 norm of the jumps across the interfaces

multiplier spaces for the mortar finite element method in three dimensions we consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions on globally or locally quasi uniform meshes a set of abstract conditions is given for the multiplier spaces which are sufficient to guarantee a stable and convergent mortar approximation three examples of multipliers satisfying these conditions are presented the first one is a dual basis example while the remaining two are based on finite volumes finally the results of computational examples illustrating the theory are reported

overlapping nonmatching grid mortar element methods for elliptic problems in the first part of the paper we introduce an overlapping mortar finite element method for solving two dimensional elliptic problems discretized on overlapping nonmatching grids we prove an optimal error bound and estimate the condition numbers of certain overlapping schwarz preconditioned systems for the two subdomain case we show that the error bound is independent of the size of the overlap and the ratio of the mesh parameters in the second part we introduce three additive schwarz preconditioned conjugate gradient algorithms based on the trivial and harmonic extensions we provide estimates for the spectral bounds on the condition numbers of the preconditioned operators we show that although the error bound is independent of the size of the overlap the condition number does depend on it numerical examples are presented to support our theory

inf sup conditions for the mortar spectral element discretization of the stokes problem summary the aim of this paper is to prove some babuska brezzi type conditions which are involved in the mortar spectral element discretization of the stokes problem for several cases of nonconforming domain decompositions

a bddc method for mortar discretizations using a transformation of basis a bddc balancing domain decomposition by constraints method is developed for elliptic equations with discontinuous coefficients discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions the coarse component of the preconditioner is defined in terms of one mortar constraint for each edge face which is the intersection of the boundaries of a pair of subdomains a condition number bound of the form c max i 1 log h i h i 2 is established under certain assumptions on the geometrically nonconforming subdomain partition in the three dimensional case here h i and h i are the subdomain diameters and the mesh sizes respectively in the geometrically conforming case and the geometrically nonconforming cases in two dimensions no assumptions on the subdomain partition are required this bddc preconditioner is also shown to be closely related to the neumann dirichlet version of the feti dp algorithm the results are illustrated by numerical experiments which confirm the theoretical results

a stabilized domain decomposition method with nonmatching grids for the stokes problem in three dimensions we present and study a nonconforming domain decomposition method for the discretization of the three dimensional stokes problem in the velocity pressure formulation the approximation is based on some local mixed finite elements for nonmatching tetrahedral grids the aim pursued is a systematic construction of the mortared discrete velocity space the pressure being not subjected to any matching constraints across the interfaces using the bubble stabilization techniques applied in brezzi and marini s paper to the three fields method math comp 70 2001 pp 911 934 allows us to define an algorithm which is easy to implement the numerical analysis relies on the pressure splitting argument of boland and nicolaides and allows us to establish an inf sup condition with a constant that does not depend on the mesh size or on the total number of the subdomains then by the berger scott strang lemma written down for our saddle point system we derive optimal accuracy results

a cal v cycle multigrid approach for mortar finite elements mortar methods based on dual lagrange multipliers provide a flexible tool for the numerical approximation of partial differential equations the associated finite element spaces are in general nonconforming and nonnested optimal multigrid results have previously been established for cal w cycle and the variable cal v cycle multigrid methods in this paper we introduce a new multigrid method based on a nested sequence of modified mortar spaces for which we can establish that the cal v cycle with one smoothing step has contraction numbers uniformly bounded away from one to obtain nested mortar spaces we apply a product form of certain corrections at the interfaces numerical results demonstrate the efficiency of the resulting multigrid solver

a new domain decomposition method with overlapping patches for ultrascale simulations application to biological flows we address the failure in scalability of large scale parallel simulations that are based on semi implicit time stepping and hence on the solution of linear systems on thousands of processors we develop a general algorithmic framework based on domain decomposition that removes the scalability limitations and leads to optimal allocation of available computational resources it is a non intrusive approach as it does not require modification of existing codes specifically we present here a two stage domain decomposition method for the navier stokes equations that combines features of discontinuous and continuous galerkin formulations at the first stage the domain is subdivided into overlapping patches and within each patch a c 0 spectral element discretization second stage is employed solution within each patch is obtained separately by applying an efficient parallel solver proper inter patch boundary conditions are developed to provide solution continuity while a multilevel communicating interface mci is developed to provide efficient communication between the non overlapping groups of processors of each patch the overall strong scaling of the method depends on the number of patches and on the scalability of the standard solver within each patch this dual path to scalability provides great flexibility in balancing accuracy with parallel efficiency the accuracy of the method has been evaluated in solutions of steady and unsteady 3d flow problems including blood flow in the human intracranial arterial tree benchmarks on bluegene p cray xt5 and sun constellation linux cluster have demonstrated good performance on up to 96 000 cores solving up to 8 21b degrees of freedom in unsteady flow problem the proposed method is general and can be potentially used with other discretization methods or in other applications

the hp mortar finite element method for the mixed elasticity and stokes problems abstract the motivation of this work is to apply the hp version of the mortar finite element method to the nearly incompressible elasticity model formulated as a mixed displacement pressure problem as well as to stokes equations in primal velocity pressure variables within each subdomain the local approximation is designed using div stable hp mixed finite elements the displacement is computed in a mortared space while the pressure is not subjected to any constraints across the interfaces by a boland nicolaides argument we prove that the discrete saddle point problem satisfies a babuska brezzi inf sup condition the inf sup constant is optimal in the sense that it depends only on the local to the subdomains characteristics of the mixed finite elements and in particular it does not increase with the total number of the subdomains the consequences that we are aware of of such an important result are twofold 1 the numerical analysis of the approximability properties of the hp mortar discretization for the mixed elasticity problem allows us to derive an asymptotic rate of convergence that is optimal up to log p in the displacement this is addressed in the present contribution 2 when the mortar discrete problem is inverted by substructured iterative methods based on krylov subspaces with block preconditioners in view of the results for conforming finite elements 1 the condition number of the solver should grow logarithmically on p h and not depend on the total number of the subdomains

two level schwarz algorithms with overlapping subregions for mortar finite elements preconditioned conjugate gradient methods based on two level overlapping schwarz methods often perform quite well such a preconditioner combines a coarse space solver with local components which are defined in terms of subregions that form an overlapping covering of the region on which the elliptic problem is defined precise bounds on the rate of convergence of such iterative methods have previously been obtained in the case of conforming lower order and spectral finite elements as well as in a number of other cases in this paper this domain decomposition algorithm and analysis are extended to mortar finite elements it is established that the condition number of the relevant iteration operator is independent of the number of subregions and varies with the relative overlap between neighboring subregions linearly as in the conforming cases previously considered

an adaptive bddc algorithm in variational form for mortar discretizations abstract a balancing domain decomposition by constraints bddc algorithm with adaptive primal constraints in variational form is introduced and analyzed for high order mortar discretization of two dimensional elliptic problems with high varying and random coefficients some vector valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm and the coarse space is formed by using a transformation operator on each interface compared with the adaptive bddc algorithms for conforming galerkin approximations our algorithm is simpler because there have not any continuity constraints at subdomain vertices in the mortar method involved in this paper the condition number of the preconditioned system is proved to be bounded above by a user defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain and independent of the mesh size and the contrast of the given coefficients numerical results show the robustness and efficiency of the algorithm for various model problems

finite element approximation of multi scale elliptic problems using patches of elements in this paper we present a method for the numerical solution of elliptic problems with multi scale data using multiple levels of not necessarily nested grids the method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming this paper provides proofs of the results published earlier see c r acad sci paris ser i 337 2003 679 684 gives a generalization of the latter to more than two domains and contains extensive numerical illustrations new results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened cauchy buniakowski schwarz inequality are presented

a numerical study of feti algorithms for mortar finite element methods the finite element tearing and interconnecting feti method is an iterative substructuring method using lagrange multipliers to enforce the continuity of the finite element solution across the subdomain interface mortar finite elements are nonconforming finite elements that allow for a geometrically nonconforming decomposition of the computational domain into subregions and at the same time for the optimal coupling of different variational approximations in different subregions we present a numerical study of feti algorithms for elliptic self adjoint equations discretized by mortar finite elements several preconditioners which have been successful for the case of conforming finite elements are considered we compare the performance of our algorithms when applied to classical mortar elements and to a new family of biorthogonal mortar elements and we discuss the differences between enforcing mortar conditions instead of continuity conditions for the case of matching nodes across the interface our experiments are carried out for both two and three dimensional problems and include a study of the relative costs of applying different preconditioners for mortar elements

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

substructuring preconditioners for the three fields domain decomposition method we study a class of preconditioners based on substructuring for the discrete steklov poincare operator arising in the three fields formulation of domain decomposition in two dimensions under extremely general assumptions on the discretization spaces involved an upper bound is provided on the condition number of the preconditioned system which is shown to grow at most as log h h 2 h and h denoting respectively the diameter and the discretization mesh size of the subdomains extensive numerical tests performed on both a plain and a stabilized version of the method confirm the optimality of such bound

szemeredi trotter type theorems in dimension 3 abstract we estimate the number of incidences in a configuration of m lines and n points in dimension 3 the main term is m n 1 3 if we work over the real or complex numbers but m n 2 5 over finite fields

a hybrid multi time step framework for pore scale and continuum scale modeling of solute transport in porous media abstract in this paper we propose a computational framework which is based on a domain decomposition technique to employ both finite element method which is a popular continuum modeling approach and lattice boltzmann method which is a popular pore scale modeling approach in the same computational domain to bridge the gap across the disparate length and time scales we first propose a new method to enforce continuum scale boundary conditions i e dirichlet and neumann boundary conditions onto the numerical solution from the lattice boltzmann method this method is based on maximization of entropy and preserve the non negativity of discrete distributions under the lattice boltzmann method the proposed computational framework allows different grid sizes orders of interpolation and time steps in different subdomains this allows for different desired resolutions in the numerical solution in different subdomains through numerical experiments the effect of grid and time step refinement disparity of time steps in different subdomains domain partitioning and the number of iteration steps on the accuracy and rate of convergence of the proposed methodology are studied finally to showcase the performance of this framework in porous media applications we use it to simulate the dissolution of calcium carbonate in a porous structure

summing logarithmic expansions for singularly perturbed eigenvalue problems strong localized perturbations of linear and nonlinear eigenvalue problems in a bounded two dimensional domain d are considered the effects on an eigenvalue lambda 0 of the lapla cian and on the fold point lambda c0 of a nonlinear eigenvalue problem of removing a small subdomain d epsilon of radius epsilon from d and imposing a condition on the boundary of the resulting hole are determined using the method of matched asymptotic expansions it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start with infinite series in powers of 1 log epsilon d kappa here d kappa is a constant that depends on the shape of d epsilon and on the precise form of the boundary condition on the hole in each case it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole this related problem is not stiff and can be solved numerically

a geometric multigrid preconditioning strategy for dpg system matrices abstract the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2010 2011 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes a key question that has not yet been answered in general though there are some results for poisson e g is how best to precondition the dpg system matrix so that iterative solvers may be used to allow solution of large scale problems in this paper we detail a strategy for preconditioning the dpg system matrix using geometric multigrid which we have implemented as part of camellia roberts 2014 2016 and demonstrate through numerical experiments its effectiveness in the context of several variational formulations we observe that in some of our experiments the behavior of the preconditioner is closely tied to the discrete test space enrichment we include experiments involving adaptive meshes with hanging nodes for lid driven cavity flow demonstrating that the preconditioners can be applied in the context of challenging problems we also include a scalability study demonstrating that the approach and our implementation scales well to many mpi ranks

additive schwarz methods for elliptic mortar finite element problems two variants of the additive schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed the methods are defined on subdomains without overlap and they use special coarse spaces resulting in algorithms that are well suited for parallel computation the condition number estimate for the preconditioned system in each method is proportional to the ratio h h where h and h are the mesh sizes and it is independent of discontinuous jumps of the coefficients for one of the methods presented the choice of the mortar nonmortar side is independent of the coefficients

uniform convergence and schwarz method for the mortar projection nonconforming and mixed element methods for nonselfadjoint and indefinite problems abstract in this paper two mortar versions of the so called projection nonconforming and the mixed element methods are proposed respectively for nonselfadjoint and indefinite second order elliptic problems it is proven that the mortar mixed element method is equivalent to the mortar projection nonconforming element method based on this equivalence the existence uniqueness and uniform convergence of the solution for mortar mixed element method are shown only under minimal regularity assumption meanwhile the optimal error estimate is obtained under certain regularity assumption furthermore an additive schwarz preconditioning method is proposed for solving the discrete problem and the nearly optimal convergence rate for the preconditioned gmres method is proven under minimal regularity assumption finally the practical implementation of the method is adderssed and numerical experiments are presented

maximum norm stability of difference schemes for parabolic equations on overset nonmatching space time grids in this paper theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space time grids we consider parabolic equations containing a linear reaction term on a space time domain 0 t which is decomposed into an overlapping collection of cylindrical subregions of the form l 0 t for l 1 p each of the space time domains l 0 t are assumed to be independently grided in parallel according to the local geometry and space time regularity of the solution yielding space time grids with mesh parameters hl and l in particular the different space time grids need not match on the regions of overlap and the time steps l can differ from one grid to the next we discretize the parabolic equation on each local grid by employing an explicit or implicit scheme in time and a finite difference scheme in space satisfying a discrete maximum principle the local discretizations are coupled together without the use of lagrange multipliers by requiring the boundary values on each space time grid to match a suitable interpolation of the solution on adjacent grids the resulting global discretization yields a large system of coupled equations which can be solved by a parallel schwarz iterative procedure requiring some communication between adjacent subregions our analysis employs a contraction mapping argument applications of the results are briefly indicated for reaction diffusion equations with contractive terms and heterogeneous hyperbolic parabolic approximations of parabolic equations

a matrix free chimera approach based on dirichlet dirichlet coupling for domain composition purposes abstract a novel approach using an algebraic based chimera type domain composition method in the context of the finite element method for non matching overlapping unstructured grids is proposed in this work the scheme is based on both the transfer of information across each grid interface via dirichlet boundary conditions and a high order interpolation algorithm to obtain one global solution of the system the solution can be obtained iteratively with a convergence rate that is similar to that obtained with an analogous conformal mesh and the matrix vector operator can be computed with completely decoupled operations on both meshes furthermore the scheme can be set as a linear operator that can be fed to a matrix free efficient iterative solver such as the biconjugate gradient stabilized method several numerical examples using non matching unstructured grids that are partially and completely overlapped with different element sizes are presented assessing the precision and convergence rate of the method

on the erdős distinct distances problem in the plane in this paper we prove that a set of n points in r 2 has at least c n log n distinct distances thus obtaining the sharp exponent in a problem of erd os we follow the setup of elekes and sharir which in the spirit of the erlangen program allows us to study the problem in the group of rigid motions of the plane this converts the problem to one of point line incidences in space we introduce two new ideas in our proof in order to control points where many lines are incident we create a cell decomposition using the polynomial ham sandwich theorem this creates a dichotomy either most of the points are in the interiors of the cells in which case we immediately get sharp results or alternatively the points lie on the walls of the cells in which case they are in the zero set of a polynomial of suprisingly low degree and we may apply the algebraic method in order to control points incident to only two lines we use the ecnode polynomial of the rev george salmon to conclude that most of the lines lie on a ruled surface then we use the geometry of ruled surfaces to complete the proof

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

additive schwarz methods for the crouzeix raviart mortar finite element for elliptic problems with discontinuous coefficients in this paper we propose two variants of the additive schwarz method for the approximation of second order elliptic boundary value problems with discontinuous coefficients on nonmatching grids using the lowest order crouzeix raviart element for the discretization in each subdomain the overall discretization is based on the mortar technique for coupling nonmatching grids the convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements the condition number bound for the preconditioned systems is independent of the jumps of the coefficient and depend linearly on the ratio between the subdomain size and the mesh size the performance of the methods is illustrated by some numerical results

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

a spectral element methodology tuned to parallel implementations abstract we present in this paper a modification of the conforming spectral element method for the approximation of the poisson equation this new procedure allows the implementation on a parallel machine in an easier and faster way the resulting method is nonconforming but the numerical analysis shows that the error between the exact solution and the discrete solution is of the same asymptotic order as the error of the original conforming method

domain decomposition methods for mortar finite element discretizations of plate problems mortar finite element methods for clamped plate problems are discussed in the subdomains conforming the reduced hsieh clough tocher hct macroelement defined on nonmatching triangulations is utilized two additive schwarz methods for solving the system of linear equations arising from this discretization are designed and analyzed the condition numbers of the preconditioned systems are almost optimal i e they are independent of the number of subdomains and grow only logarithmically with respect to the mesh parameters

two scale dirichlet neumann preconditioners for elastic problems with boundary refinements abstract the present work deals with the efficient resolution of elastostatics problems on domains with boundary refinements the proposed approach separates the boundary refinements from the interior of the domain by the mortar method and uses dirichlet neumann preconditioners to solve the corresponding algebraic system we prove that the simplest dirichlet neumann algorithm achieves independence of the condition number of the preconditioned system with respect to the number and the size of the small details nevertheless the situation no longer prevails when the refined boundary is clamped an enhanced preconditioner is then designed by the introduction of a coarse space to mitigate the aforementioned sensitivity some numerical tests are performed to confirm the analysis and the tools are extended by the proposition of a quasi newton method in the case of nonlinear elasticity this paper is an extended version of a work presented at the dd16 conference with proofs and complete numerical results

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

the erdős szemeredi problem on sum set and product set summary the basic theme of this paper is the fact that if a is a finite set of integers then the sum and product sets cannot both be small a precise formulation of this fact is conjecture 1 below due to erd os szemeredi e s see also el t and k t for related aspects only much weaker results or very special cases of this conjecture are presently known one approach consists of assuming the sum set a a small and then deriving that the product set aa is large using freiman s structure theorem cf n t na3 we follow the reverse route and prove that if aa c a 2 see theorem 1 a quantitative version of this phenomenon combined with the plunnecke type of inequality due to ruzsa permit us to settle completely a related conjecture in e s on the growth in k i f

information transfer between incompatible finite element meshes application to coupled thermo viscoelasticity this article is concerned with information transfer between non matching finite element meshes such a feature is not new in the literature but we focus herein on a geometric approach to transfer solution fields in order to be as application independent as possible moreover the dedicated case concerns the transfer of finite element fields defined at integration points of the meshes and allows iterative exchange of fields in both directions to do so we propose an extension to the mortar technique that fulfills these goals and that does not suffer from a high computational cost the application is the simulation of a strongly coupled thermo viscoelastic problem with phase transition solved with a partitioning technique

maximum norm analysis of overlapping nonmatching grid discretizations of elliptic equations in this paper we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations we consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid the grid points on the subdomain boundaries need not match the grid points from adjacent subdomains to obtain a global finite difference discretization of the elliptic problem we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain by interpolating the discrete solution on adjacent subdomains if the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains we describe also how the resulting linear system can be solved iteratively by a parallel schwarz alternating method or a schwarz preconditioned krylov subspace iterative method several numerical results are included to support the theory

deforming composite grids for solving fluid structure problems we describe a mixed eulerian lagrangian approach for solving fluid structure interaction fsi problems the technique which uses deforming composite grids dcg is applied to fsi problems that couple high speed compressible flow with elastic solids the fluid and solid domains are discretized with composite overlapping grids curvilinear grids are aligned with each interface and these grids deform as the interface evolves the majority of grid points in the fluid domain generally belong to background cartesian grids which do not move during a simulation the fsi dcg approach allows large displacements of the interfaces while retaining high quality grids efficiency is obtained through the use of structured grids and cartesian grids the governing equations in the fluid and solid domains are evolved in a partitioned approach we solve the compressible euler equations in the fluid domains using a high order godunov finite volume scheme we solve the linear elastodynamic equations in the solid domains using a second order upwind scheme we develop interface approximations based on the solution of a fluid solid riemann problem that results in a stable scheme even for the difficult case of light solids coupled to heavy fluids the fsi dcg approach is verified for three problems with known solutions an elastic piston problem the superseismic shock problem and a deforming diffuser in addition a self convergence study is performed for an elastic shock hitting a fluid filled cavity the overall fsi dcg scheme is shown to be second order accurate in the max norm for smooth solutions and robust and stable for problems with discontinuous solutions for a wide range of constitutive parameters

a gradient method approach to optimization based multidisciplinary simulations and nonoverlapping domain decomposition algorithms it has been shown recently that optimization based nonoverlapping domain decomposition algorithms are connected to many well known algorithms using a gradient type iterative strategy for the optimization problem we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain in particular the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

bounding multiplicative energy by the sumset abstract we prove that the sumset or the productset of any finite set of real numbers a is at least a 4 3 e improving earlier bounds our main tool is a new upper bound on the multiplicative energy e a a

domain decomposition with nonmatching grids augmented lagrangian approach we propose and study a domain decomposition method which treats the constraint of displacement continuity at the interfaces by augmented la grangian techniques and solves the resulting problem by a parallel version of the peaceman rachford algorithm we prove that this algorithm is equivalent to the fictitious overlapping method introduced by p l lions we also prove its linear convergence independently of the discretization step h even if the finite element grids do not match at the interfaces a new preconditioner using fictitious overlapping and well adapted to three dimensional elasticity problems is also introduced and is validated on several numerical examples

shout options a framework for pricing contracts which can be modified by the investor a shout option may be broadly defined as a financial contract which can be modified by the holder according to specified rules in a simple example the holder could have the right to set the strike of an option equal to the current value of the underlying asset in such a case the holder effectively has the right to select when to take ownership of an at the money option more generally the holder could have multiple rights along these lines in some cases with a limit placed on the number of rights which may be exercised within a given time period e g four times per year the value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems this paper describes a general framework for the valuation of complex types of shout options numerical issues related to interpolation and choice of timestepping method are considered in detail some illustrative examples are provided

a parabolic variational inequality arising from the valuation of strike reset options abstract a strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder the pricing model of the option can be formulated as a one dimensional parabolic variational inequality or equivalently a free boundary problem where the free boundary just corresponds to the optimal reset strategy adopted by the holder of the option this paper is concerned with the theoretical analysis of the model the existence and uniqueness of the solution are established furthermore we study properties of the free boundary the monotonicity and c smoothness of the free boundary are proven in some situations

a high order accurate scheme for maxwell s equations with a generalized dispersive material model abstract a high order accurate scheme for solving the time domain maxwell s equations with a generalized dispersive material model is described the equations for the electric field are solved in second order form and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations ade numerical methods are developed for both second order and fourth order accuracy in space and time the equations are discretized using finite differences and advanced in time with a single stage three level space time scheme which remains stable up to the usual explicit cfl restriction as proven using mode analysis because the equations are treated in their second order form there is no need for grid staggering and instead a collocated grid is used composite overlapping grids are used to treat complex geometries with boundary conforming grids and a high order upwind dissipation is added to ensure robust and stable approximations on overlapping grids numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes

multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions multilevel schwarz methods are developed for a conforming finite element approximation of second order elliptic problems we focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions we establish a condition number estimate for the iterative operator which is independent of the coefficients and grows at most as the square of the number of levels we also characterize a class of distributions of the coefficients called quasi monotone for which the weighted l 2 projection is stable and for which we can use the standard piecewise linear functions as a coarse space in this case we obtain optimal methods i e bounds which are independent of the number of levels and subregions we also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas we consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods

new sum product type estimates over finite fields abstract let f be a field with positive odd characteristic p we prove a variety of new sum product type estimates over f they are derived from the theorem that the number of incidences between m points and n planes in the projective three space p g 3 f with m n o p 2 is o m n k m where k denotes the maximum number of collinear planes the main result is a significant improvement of the state of the art sum product inequality over fields with positive characteristic namely that 1 a a a a a 1 1 5 for any a such that a p 5 8

uniform hp convergence results for the mortar finite element method the mortar finite element is an example of a non conforming method which can be used to decompose and re compose a domain into sub domains without requiring compatibility between the meshes on the separate components we obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used without assuming quasiuniformity for the meshes our results establish that the method is optimal when non quasiuniform h or hp methods are used such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain we also give an error estimate for when the p version is used numerical results for hp and hp mortar fems show that these methods behave as well as conforming fems an hp extension theorem is also proved

a finite element method with overlapping meshes for free boundary axisymmetric plasma equilibria in realistic geometries existing finite element implementations for the computation of free boundary axisymmetric plasma equilibria approximate the unknown poloidal flux function by standard lowest order continuous finite elements with discontinuous gradients as a consequence the location of critical points of the poloidal flux that are of paramount importance in tokamak engineering is constrained to nodes of the mesh leading to undesired jumps in transient problems moreover recent numerical results for the self consistent coupling of equilibrium with resistive diffusion and transport suggest the necessity of higher regularity when approximating the flux map in this work we propose a mortar element method that employs two overlapping meshes one mesh with cartesian quadrilaterals covers the vacuum chamber domain accessible by the plasma and one mesh with triangles discretizes the region outside the two meshes overlap in a narrow region this approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the domain covered by the plasma while preserving accurate meshing of the geometric details outside this region the continuity of the numerical solution in the region of overlap is weakly enforced by a mortar like mapping

bps preconditioners for a weak galerkin finite element method for 2d diffusion problems with strongly discontinuous coefficients abstract we analyze two bps preconditioners for a weak galerkin wg finite element method for 2d diffusion equations with strongly discontinuous coefficients the first preconditioner uses nonconforming linear elements on the coarse mesh and the corresponding condition number is bounded from above by c 1 ln h h 3 with c independent of the coefficients the second one uses h 1 conforming linear elements on the coarse mesh and the corresponding condition number is bounded from above by c 1 ln h h 2 with c depending on the coefficients in addition we construct and analyze a preconditioner for the sub problems encountered in the procedure of applying the two preconditioners the condition number of the preconditioned system is bounded from above by c 1 ln h h 2 with c independent of the coefficients finally we carry out some numerical experiments to verify our theoretical results

infinite reload options pricing and analysis infinite reload options allow the user to exercise his reload right as often as he chooses during the lifetime of the contract each time a reload occurs the owner receives new options where the strike price is set to the current stock price we consider a modified version of the infinite reload option contract where the strike price of the new options received by the owner is increased by a certain percentage we refer to this new contract as an increased reload option the pricing problem for this modified contract is characterized as an impulse control problem resulting in a hamilton jacobi bellman equation we use fully implicit timestepping and prove that the discretized equations are monotone stable and consistent implying convergence to the viscosity solution we also derive a globally convergent iterative method for solving the non linear discrete equations numerical examples show that both the exercise policy and the option value are very sensitive to the percentage increase in the reload strike

a fast solver for navier stokes equations in the lamina regime using mortar finite element and boundary element methods a multistep method is used to solve the navier stokes equations formulated with vorticity and stream function each time step is decomposed into three stages a convection step using the character

the lifting of polynomial traces revisited we construct a lifting operator of polynomial traces on an interval that is stable in appropriate sobolev norms next we extend this result to the case of traces vanishing at the endpoints of the interval this has two applications the interpolation of polynomial spaces and the evaluation by discrete formulas of fractional order sobolev norms on polynomials

the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

a multigrid method based on the unconstrained product space for mortar finite element discretizations the mortar finite element method allows the coupling of different discretizations across subregion boundaries in the original mortar approach the lagrange multiplier space enforcing a weak continuity condition at the interfaces is defined as a modified finite element trace space here we present a new approach where the lagrange multiplier space is replaced by a dual space without losing the optimality of the a priori bounds we introduce new dual spaces in 2d and 3d using the biorthogonality between the nodal basis functions of this lagrange multiplier space and a finite element trace space we derive an equivalent symmetric positive definite variational problem defined on the unconstrained product space the introduction of this formulation is based on a local elimination process for the lagrange multiplier this equivalent approach is the starting point for the efficient iterative solution by a multigrid method to obtain level independent convergence rates for the cw cycle we have to define suitable level dependent bilinear forms and transfer operators numerical results illustrate the performance of our multigrid method in 2d and 3d

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

a penalty method for american options with jump diffusion processes the fair price for an american option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem we develop an implicit discretization method for pricing such american options the jump diffusion correlation integral term is computed using an iterative method coupled with an fft while the american constraint is imposed by using a penalty method we derive sufficient conditions for global convergence of the discrete penalized equations at each timestep finally we present numerical tests which illustrate such convergence