analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

impact of element level static condensation on iterative solver performance this paper provides theoretical estimates that quantify and clarify the savings associated to the use of element level static condensation as a first step of an iterative solver these estimates are verified numerically the numerical evidence shows that static condensation at the element level is beneficial for higher order methods for lower order methods or when the number of iterations required for convergence is low the setup cost of the elimination as well as its implementation may offset the benefits obtained during the iteration process however as the iteration count e g above 50 or the polynomial order e g above cubics grows the benefits of element level static condensation are significant

weak dynamic programming principle for viscosity solutions we prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control stopping problems which avoids the technical difficulties related to the measurable selection argument in the markov case our result is tailor made for the derivation of the dynamic programming equation in the sense of viscosity solutions

overlapping schwarz preconditioners for indefinite time harmonic maxwell equations time harmonic maxwell equations in lossless media lead to a second order differential equation for the electric field involving a differential operator that is neither elliptic nor definite a galerkin method using nedelec spaces can be employed to get approximate solutions numerically the problem of preconditioning the indefinite matrix arising from this method is discussed here specifically two overlapping schwarz methods will be shown to yield uniform preconditioners

european option pricing with transaction costs the authors consider the problem of pricing european options in a market model similar to the black scholes one except that proportional transaction charges are levied on all sales and purchases o

an upwind numerical approach for an american and european option pricing model the numerical solution of several mathematical models arising in financial economics for the valuation of both european and american call options on different types of assets is considered all the models are based on the black scholes partial differential equation in the case of european options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented when treating with american options an additional inequality constraint leads to a discretized linear complementarity problem in each case the numerical approximation of option values is computed by means of optimization algorithms in particular uzawa s method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics

multigrid methods for two dimensional maxwell s equations on graded meshes in this paper we investigate the numerical solution for two dimensional maxwell s equations on graded meshes the approach is based on the hodge decomposition the solution u of maxwell s equations is approximated by solving standard second order elliptic problems quasi optimal error estimates for both u and i u in the l 2 norm are obtained on graded meshes we prove the uniform convergence of the w cycle and full multigrid algorithms for the resulting discrete problem the performance of these methods is illustrated by numerical results

compatible coarse nodal and edge elements through energy functionals we propose new algorithms for the setup phase of algebraic multigrid amg solvers for linear systems coming from edge element discretization the construction of coarse levels is performed by solving an optimization problem with a lagrange multiplier method we minimize the energy of coarse bases under a constraint linking coarse nodal and edge element bases on structured meshes the resulting amg method and the geometric multigrid method behave similarly as preconditioners on unstructured meshes the convergence rate of our method compares favorably with the amg method of reitzinger and schoberl

a mortar edge element method with nearly optimal convergence for three dimensional maxwell s equations in this paper we are concerned with mortar edge element methods for solving three dimensional maxwell s equations a new type of lagrange multiplier space is introduced to impose the weak continuity of the tangential components of the edge element solutions across the interfaces between neighboring subdomains the mortar edge element method is shown to have nearly optimal convergence under some natural regularity assumptions when nested triangulations are assumed on the interfaces a generalized edge element interpolation is introduced which plays a crucial role in establishing the nearly optimal convergence the theoretically predicted convergence is confirmed by numerical experiments

convergent finite element discretization of the multi fluid nonstationary incompressible magnetohydrodynamics equations we propose a convergent implicit stabilized finite element discretization of the nonstationary incompressible magnetohydrodynamics equations with variable density viscosity and electric conductivity the discretization satisfies a discrete energy law and a discrete maximum principle for the positive density and iterates converge to weak solutions of the limiting problem for vanishing discretization parameters a simple fixed point scheme together with an appropriate stopping criterion is proposed which decouples the computation of density velocity and magnetic field and inherits the above properties provided a mild mesh constraint holds computational studies are provided

option pricing with a direct adaptive sparse grid approach we present an adaptive sparse grid algorithm for the solution of the black scholes equation for option pricing using the finite element method sparse grids enable us to deal with higher dimensional problems better than full grids in contrast to common approaches that are based on the combination technique which combines different solutions on anisotropic coarse full grids the direct sparse grid approach allows for local adaptive refinement when dealing with non smooth payoff functions this reduces the computational effort significantly in this paper we introduce the spatially adaptive discretization of the black scholes equation with sparse grids and describe the algorithmic structure of the numerical solver we present several strategies for adaptive refinement evaluate them for different dimensionalities and demonstrate their performance showing numerical results

multigrid in a weighted space arising from axisymmetric electromagnetics consider the space of two dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable this space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates we prove that if the original three dimensional domain is convex then the multigrid v cycle applied to the inner product in this space converges provided certain modern smoothers are used for the convergence analysis we first prove several intermediate results e g the approximation properties of a commuting projector in weighted norms and a superconvergence estimate for a dual mixed method in weighted spaces the uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments

robust algebraic multilevel preconditioning in h curl and h div abstract an algebraic multilevel iteration method for solving a system of linear algebraic equations arising in h curl and h div spaces is presented the algorithm is developed for the discrete problem obtained by using the space of lowest order nedelec and raviart thomas nedelec elements the theoretical analysis of the method is based only on some algebraic sequences and generalized eigenvalues of local element wise problems in the hierarchical basis framework explicit recursion formulas are derived to compute the element matrices and the constant which measures the quality of the space splitting at any given level it is proved that the proposed method is robust with respect to the problem parameters and is of optimal order complexity supporting numerical results including the case when the parameters have jumps are also presented

robust multigrid method for the planar linear elasticity problems we consider the solution of the system of equations that arise from the higher order conforming finite element scott vogelius element discretizations of the boundary value problems associated with the differential operator 2 2 div where and are nonzero parameters robust multigrid method is constructed i e the convergence rate of multigrid method is optimal with respect to the mesh size the number of levels and weights on the two terms in the aforementioned differential operator

mixed finite element methods for incompressible flow stationary navier stokes equations in z cai c tong p s vassilevski and c wang numer methods partial differential equations to appear the authors developed and analyzed a mixed finite element method for the stationary stokes equations based on the pseudostress velocity formulation the pseudostress and the velocity are approximated by a stable pair of finite elements raviart thomas elements of index k geq0 and discontinuous piecewise polynomials of degree k geq0 respectively this paper extends the method to the stationary incompressible navier stokes equations under appropriate assumptions we show that the pseudostress velocity formulation of the navier stokes equation and its discrete counterpart have branches of nonsingular solutions and error estimates of the mixed finite element approximations are established as well

a discontinuous galerkin method on refined meshes for the two dimensional time harmonic maxwell equations in composite materials in this paper a discontinuous galerkin method for the two dimensional time harmonic maxwell equations in composite materials is presented the divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized due to an appropriate mesh refinement near exterior and interior corners the singular behaviour of the electromagnetic field is taken into account optimal error estimates in a discrete energy norm and in the l 2 norm are proved in the case where the exact solution is singular

algebraic multilevel iteration method for lowest order raviart thomas space and applications an optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems that have their weak formulation in the space h div is presented the algorithm is developed for the discrete problem obtained by using the lowest order raviart thomas space the method is theoretically analyzed and supporting numerical examples are presented furthermore as a particular application the algorithm is used for the solution of the discrete minimization problem which arises in the functional type a posteriori error estimates for the discontinuous galerkin approximation of elliptic problems copyright 2011 john wiley sons ltd

mixed finite element methods for stationary incompressible magneto hydrodynamics a new mixed variational formulation of the equations of stationary incompressible magneto hydrodynamics is introduced and analyzed the formulation is based on curl conforming sobolev spaces for the magnetic variables and is shown to be well posed in possibly non convex lipschitz polyhedra a finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf sup stable velocity pressure space pairs and the magnetic ones by a mixed approach using nedelec s elements of the first kind an error analysis is carried out that shows that the proposed finite element approximation leads to quasi optimal error bounds in the mesh size

an iterative substructuring method for raviart thomas vector fields in three dimensions the iterative substructuring methods also known as schur complement methods form one of two important families of domain decomposition algorithms they are based on a partitioning of a given region on which the partial differential equation is defined into nonoverlapping substructures the preconditioners of these conjugate gradient methods are then given in terms of local problems defined on individual substructures and pairs of substructures and in addition a global problem of low dimension an iterative method of this kind is introduced for the lowest order raviart thomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure the theoretical bounds are confirmed by a series of numerical experiments

multilevel iterative solvers for the edge finite element solution of the 3d maxwell equation in the edge vector finite element solution of the frequency domain maxwell equations the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers we extend the approach of r hiptmair multigrid method for maxwell s equations siam j numer anal 36 1 1999 204 225 and using domain decomposition ideas construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements the new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain maxwell equations the solver can be seen as a variable preconditioner and thus accelerated by krylov subspace techniques e g gcr or fgmres we demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity

a multigrid preconditioner for the mixed formulation of linear plane elasticity in this paper we develop a multigrid preconditioner for the discrete system of linear equations that results from the mixed formulation of the linear plane elasticity problem using the arnold winther elements this in turn can be reduced to the problem of finding a multigrid preconditioner for the form cdot cdot vdiv cdot vdiv cdot in the symmetric matrix space resulting from arnold winther elements since the form is not uniformly elliptic a helmholtz type decomposition is essential the arnold winther finite element space gives rise to nonnested multilevel spaces adding difficulty to the analysis we prove that for the variable v cycle multigrid preconditioner the condition number of the preconditioned system is independent of the number of levels the results of numerical experiments are also presented

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a nonoverlapping domain decomposition method for maxwell s equations in three dimensions in this paper we propose a nonoverlapping domain decomposition method for solving the three dimensional maxwell equations based on the edge element discretization for the schur complement system on the interface we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping subdomains it is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients

geometric multigrid for darcy and brinkman models of flows in highly heterogeneous porous media we apply geometric multigrid methods for the finite element approximation of flow problems governed by darcy and brinkman systems used in modeling highly heterogeneous porous media the method is based on divergence conforming discontinuous galerkin methods and overlapping patch based domain decomposition smoothers we show in benchmark experiments that the method is robust with respect to mesh size and contrast of permeability for highly heterogeneous media

numerical solution of generalized black scholes model abstract this paper presents a numerical scheme that approximates the option prices for different option styles governed by the generalized black scholes equation in its degenerate form the proposed method uses the hodie scheme in the spacial direction and the two step backward differentiation formula in the temporal direction it is proved that the method has second order convergence in space as well as in time numerical experiments validate the theoretical results

convergence of a mixed method for a semi stationary compressible stokes system we propose and analyze a finite element method for a semi stationary stokes system modeling compressible fluid flow subject to a navier slip boundary condition the velocity momentum equation is approximated by a mixed finite element method using the lowest order nedelec spaces of the first kind the continuity equation is approximated by a standard piecewise constant upwind discontinuous galerkin scheme our main result states that the numerical method converges to a weak solution the convergence proof consists of two main steps i to establish strong spatial compactness of the velocity field which is intricate since the element spaces are only div or curl conforming ii to prove that the discontinuous galerkin approximations converge strongly which is required in view of the nonlinear pressure function tools involved in the analysis include a higher integrability estimate for the discontinuous galerkin approximations a discrete equation for the effective viscous flux and various renormalized formulations of the discontinuous galerkin scheme

error analysis of a finite element integral equation scheme for approximating the time harmonic maxwell system in 1996 hazard and lenoir suggested a variational formulation of maxwell s equations using an overlapping integral equation and volume representation of the solution siam j math anal 27 1996 pp 1597 1630 they suggested a numerical scheme based on this approach but no error analysis was provided in this paper we provide a convergence analysis of an edge finite element scheme for the method the analysis uses the theory of collectively compact operators its novelty is that a perturbation argument is needed to obtain error estimates for the solution of the discrete problem that is best suited for implementation

first order system least squares fosls for spatial linear elasticity pure traction this paper develops first order system least squares fosls functionals for solving the pure traction problem in three dimensional linear elasticity it is a direct extension of an earlier paper on planar elasticity z cai t a manteuffel s f mccormick and s v parter siam j numer anal 35 1998 pp 320 335 two functionals are developed one involving l2 norms of the first order system and the other involving dual norms these functionals are shown to be equivalent to appropriate product sobolev norms uniformly in the poisson ratio these results imply that standard finite element discretization and iterative solver techniques can be applied to obtain performance that is optimal even as the material nears the incompressible limit

a mixed method for axisymmetric div curl systems we present a mixed method for a three dimensional axisymmetric div curl system reduced to a two dimensional computational domain via cylindrical coordinates we show that when the meridian axisymmetric maxwell problem is approximated by a mixed method using the lowest order nedelec elements for the vector variable and linear elements for the lagrange multiplier one obtains optimal error estimates in certain weighted sobolev norms the main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

a finite element framework for some mimetic finite difference discretizations in this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified nedelec raviart thomas finite element methods for model problems in h curl and h div this provides a simple and transparent way to analyze such mimetic finite difference discretizations using the well known results from finite element theory the finite element framework that we develop is also crucial for the design of efficient multigrid methods for mimetic finite difference discretizations since it allows us to use canonical inter grid transfer operators arising from the finite element framework we provide special local fourier analysis and numerical results to demonstrate the efficiency of such multigrid methods

biological population management based on a hamilton jacobi bellman equation with boundary blow up establishment of a long term biological population management policy requires balancing its cost and utility disutility we approach this issue from the viewpoint of stochastic control as an effici

a robust multigrid method for discontinuous galerkin discretizations of stokes and linear elasticity equations we consider multigrid methods for discontinuous galerkin h text div omega h div conforming discretizations of the stokes and linear elasticity equations we analyze variable v cycle and w cycle multigrid methods with nonnested bilinear forms we prove that these algorithms are optimal and robust i e their convergence rates are independent of the mesh size and also of the material parameters such as the poisson ratio numerical experiments are conducted that further confirm the theoretical results

auxiliary space preconditioning in h 0 curl ω we adapt the principle of auxiliary space preconditioning as presented in j xu the auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids computing 56 1996 pp 215 235 to h curl elliptic variational problems discretized by means of edge elements the focus is on theoretical analysis within the abstract framework of subspace correction employing a helmholtz type splitting of edge element vector fields we can establish asymptotic h uniform optimality of the preconditioner defined by our auxiliary space method

shadow prices and well posedness in the problem of optimal investment and consumption with transaction costs we revisit the optimal investment and consumption model of davis and norman math oper res 15 1990 pp 676 713 and shreve and soner ann appl probab 4 1994 pp 609 692 following a shadow price approach similar to that of kallsen and muhle karbe ann appl probab 20 2010 pp 1341 1358 making use of the completeness of the model without transaction costs we reformulate and reduce the hamilton jacobi bellman equation for this singular stochastic control problem to a nonstandard free boundary problem for a first order ode with an integral constraint having shown that the free boundary problem has a smooth solution we use it to construct the solution of the original optimal investment consumption problem in a self contained manner and without any recourse to the dynamic programming principle furthermore we provide an explicit characterization of model parameters for which the value function is finite

analysis of finite element approximation for time dependent maxwell problems we provide an error analysis of finite element methods for solving time dependent maxwell problem using nedelec and thomas raviart elements we study the regularity of the solution and develop some new error estimates of nedelec finite elements as a result the optimal l 2 error bound for the semidiscrete scheme is obtained

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

a modified least squares mixed finite element with improved momentum balance the main goal of this contribution is to provide an improved mixed finite element for quasi incompressible linear elasticity based on a classical least squares formulation a modified weak form with displacements and stresses as process variables is derived this weak form is the basis for a finite element with an advanced fulfillment of the momentum balance and therefore with a better performance for the continuous approximation of stresses and displacements on the triangular and tetrahedral elements lowest order raviart thomas and linear standard lagrange interpolations can be used it is shown that coercivity and continuity of the resulting asymmetric bilinear form could be established with respect to appropriate norms further on details about the implementation of the least squares mixed finite elements are given and some numerical examples are presented in order to demonstrate the performance of the proposed formulation copyright 2009 john wiley sons ltd

analysis of a stabilized finite element method for stokes equations of velocity boundary condition and of pressure boundary condition abstract in this paper a new pressure stabilized finite element method is analyzed for solving the stokes equations the key feature of the method is using the curl integral instead of the standard dirichlet integral and using the pressure stabilization instead of the inf sup condition thus the method is very flexibly applicable to two problems of stokes equations in terms of velocity and pressure with either pressure dirichlet boundary condition or velocity dirichlet boundary condition for both stokes problems the finite element scheme and the finite element space of velocity are the same and one where only the finite element spaces of pressure have small differences a general analysis of stability and error estimates is developed and applications to both stokes problems are further analyzed the method covers the low regularity solution of very weak non h 1 space solution of stokes problem of pressure dirichlet boundary condition as well as higher regularity solutions of stokes problems of either pressure dirichlet boundary condition or velocity dirichlet boundary condition numerical results are presented to illustrate the performance of the method and to confirm the theoretical results

least squares for the perturbed stokes equations and the reissner mindlin plate in this paper we develop two least squares approaches for the solution of the stokes equations perturbed by a laplacian term such perturbed stokes equations arise from finite element approximations of the reissner mindlin plate both are two stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain then for the rotation itself if desired one approach uses l2 norms and the other approach uses h 1 norms to define the least squares functionals it is shown that the h 1 norm approach under general assumptions and the l2 norm approach under certain h2 regularity assumptions admit optimal performance for standard finite element discretization and either standard multigrid solution methods or preconditioners these methods do not degrade when the perturbed parameter the plate thickness approaches zero we also develop a three stage least squares method for the reissner mindlin plate which first solves for the curls of the rotation and the shear strain next for the rotation itself and then for the transverse displacement

multilevel iteration for mixed finite element systems with penalty in this paper the authors consider the solution of the discrete systems that arises when a mixed finite element approach is used to approximate the solution of second order elliptic boundary value problems by the introduction of a penalty parameter these equations can be approximated by the solution of a symmetric and positive definite penalty system on the velocity subspace iterative procedures are developed and analyzed for this penalty system based on the hierarchical basis approach as well as on the standard multigrid approach finally numerical experiments are presented that illustrate the convergence behavior suggested by the theory

a posteriori error estimates for maxwell equations maxwell equations are posed as variational boundary value problems in the function space h curl and are discretized by nedelec finite elements in beck et al 2000 a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain in the present paper we prove the reliability of that error estimator on lipschitz domains the key is to establish new error estimates for the commuting quasi interpolation operators recently introduced in j schoberl commuting quasi interpolation operators for mixed finite elements similar estimates are required for additive schwarz preconditioning to incorporate boundary conditions we establish a new extension result

mixed methods for stationary navier stokes equations based on pseudostress pressure velocity formulation in this paper we develop and analyze mixed finite element methods for the stokes and navier stokes equations our mixed method is based on the pseudostress pressure velocity formulation the pseudostress is approximated by the raviart thomas brezzi douglas marini or brezzi douglasfortin marini elements the pressure and the velocity by piecewise discontinuous polynomials of appropriate degree it is shown that these sets of finite elements are stable and yield optimal accuracy for the stokes problem for the pseudostress pressure velocity formulation of the stationary navier stokes equations the well posedness and error estimation results are established by eliminating the pseudostress variables in the resulting algebraic system we obtain cell centered finite volume schemes for the velocity and pressure variables that preserve local balance of momentum

on the uniqueness of unbounded viscosity solutions arising in an optimal terminal wealth problem with transaction costs we study the uniqueness of viscosity solutions of a hamilton jacobi bellman equation which arises in a portfolio optimization problem in which an investor maximizes expected utility of terminal wealth in the presence of proportional transaction costs our main contribution is that the comparison theorem can be applied to prove the uniqueness of the value function in the portfolio optimization problem for logarithmic and power utility

wavelet bases in h div and h curl some years ago compactly supported divergence free wavelets have been constructed which also give rise to a stable biorthogonal wavelet splitting of h div these bases have successfully been used both in the analysis and numerical treatment of the stokes and navier stokes equations in this paper we construct stable wavelet bases for the stream function spaces h curl moreover curl free vector wavelets are constructed and analysed the relationship between h div and h curl are expressed in terms of these wavelets we obtain discrete orthogonal hodge decompositions our construction works independently of the space dimension but in terms of general assumptions on the underlying wavelet systems in l 2 omega that are used as building blocks we give concrete examples of such bases for tensor product and certain more general domains omega subset er n as an application we obtain wavelet multilevel preconditioners in h div and h curl email urban dragon ian pv cnr it keywords h div h curl stream function spaces wavelets

sparsity optimized high order finite element functions for h div on simplices this paper deals with conforming high order finite element discretizations of the vector valued function space h div in 2 and 3 dimensions a new set of hierarchic basis functions on simplices with the following two main properties is introduced when working with an affine simplicial triangulation first the divergence of the basis functions is l 2 orthogonal and secondly the l 2 inner product of the interior basis functions is sparse with respect to the polynomial degree p the construction relies on a tensor product based construction with properly weighted jacobi polynomials as well as an explicit splitting of the higher order basis functions into solenoidal and non solenoidal ones the basis is suited for fast assembling strategies the general proof of the sparsity result is done by the assistance of computer algebra software tools several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter dependent div div problem even on curved elements a tremendous improvement in condition numbers is observed the precomputed mass and stiffness matrix entries in general form are available online

a weighted adaptive least squares finite element method for the poisson boltzmann equation abstract the finite element methodology has become a standard framework for approximating the solution to the poisson boltzmann equation in many biological applications in this article we examine the numerical efficacy of least squares finite element methods for the linearized form of the equations in particular we highlight the utility of a first order form noting optimality control of the flux variables and flexibility in the formulation including the choice of elements we explore the impact of weighting and the choice of elements on conditioning and adaptive refinement in a series of numerical experiments we compare the finite element methods when applied to the problem of computing the solvation free energy for realistic molecules of varying size

two level additive preconditioners for edge element discretizations of time harmonic maxwell equations two level additive preconditioners are presented for edge element discretizations of time harmonic maxwell equations the key is to construct a special coarse mesh space which adds the kernel of the curl operator in a fine space to a coarse mesh space to solve the original problem and then uses the fine mesh space to solve the h curl elliptic problem it is shown that the generalized minimal residual gmres method applied to the preconditioned system converges uniformly provided that the coarsest mesh size is reasonably small but independent of the fine mesh size and the parameter for the coarse mesh space solver is sufficiently large numerical experiments show the efficiency of the proposed approach

estimation of local volatilities in a generalized black scholes model this paper studies a parameter estimation problem for a generalized black scholes equation which is used for option pricing in estimating the volatility function from a set of market observations we use an implicit finite difference scheme the function space parameter estimation convergence fspec is proved and numerical simulations were performed

analysis of block preconditioners for models of coupled magma mantle dynamics this article considers the iterative solution of a finite element discretization of the magma dynamics equations in simplified form the magma dynamics equations share some features of the stokes equations we therefore formulate analyze and numerically test an elman silvester and wathen type block preconditioner for magma dynamics we prove analytically and demonstrate numerically the optimality of the preconditioner the presented analysis highlights the dependence of the preconditioner on parameters in the magma dynamics equations that can affect convergence of iterative linear solvers the analysis is verified through a range of two and three dimensional numerical examples on unstructured grids from simple illustrative problems through to large problems on subduction zone like geometries the computer code to reproduce all numerical examples is freely available as supporting material

weak dynamic programming for generalized state constraints we provide a dynamic programming principle for stochastic optimal control problems with expectation constraints a weak formulation using test functions and a probabilistic relaxation of the constraint avoids restrictions related to a measurable selection but still implies the hamilton jacobi bellman equation in the viscosity sense we treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints

full wave analysis of dielectric waveguides at a given frequency new variational formulation to compute propagation constants is proposed based on it vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property convergence analysis is conducted in the framework of collectively compact operators reported theoretical results apply to a wide class of vector finite elements including two families of nedelec and their generalization the hp edge elements numerical experiments fully support theoretical estimates for convergence rates

two grid methods for maxwell eigenvalue problems two new two grid algorithms are proposed for solving the maxwell eigenvalue problem the new methods are based on the two grid methodology recently proposed by xu and zhou math comp 70 2001 pp 17 25 and further developed by hu and cheng math comp 80 2011 pp 1287 1301 for elliptic eigenvalue problems the new two grid schemes reduce the solution of the maxwell eigenvalue problem on a fine grid to one linear indefinite maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid the new schemes therefore save total computational cost the error estimates reveals that the two grid methods maintain asymptotically optimal accuracy and the numerical experiments presented confirm the theoretical results

time integration methods for finite element discretisations of the second order maxwell equation this article deals with time integration for the second order maxwell equations with possibly non zero conductivity in the context of the discontinuous galerkin finite element method dg fem and the h curl conforming fem for the spatial discretisation hierarchic h curl conforming basis functions are used up to polynomial order p 3 over tetrahedral meshes meaning fourth order convergence rate a high order polynomial basis often warrants the use of high order time integration schemes but many well known high order schemes may suffer from a severe time step stability restriction owing to the conductivity term we investigate several possible time integration methods from the point of view of accuracy stability and computational work we also carry out a numerical fourier analysis to study the dispersion and dissipation properties of the semi discrete dg fem scheme as well as the fully discrete schemes with several of the time integration methods the dispersion and dissipation properties of the spatial discretisation and those of the time integration methods are investigated separately providing additional insight into the two discretisation steps

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given

incompressible finite element methods for navier stokes equations with nonstandard boundary conditions in r 3 application des methodes a elements finis incompressibles aux equations de navier stokes avec des conditions aux limites non standards dans r 3 utilisation du potentiel vecteur de la vorticite et de la pression estimations d erreurs

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

finite horizon optimal investment with transaction costs a parabolic double obstacle problem abstract this paper concerns optimal investment problem of a crra investor who faces proportional transaction costs and finite time horizon from the angle of stochastic control it is a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies this enables us to make use of the well developed theory of obstacle problem to attack the problem the c 2 1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized

convergence of adaptive mixed finite element methods for the hodge laplacian equation without harmonic forms finite element exterior calculus feec has been developed as a systematic framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes this paper is devoted to analyzing the convergence of adaptive mixed finite element methods for the hodge laplacian equations based on feec without considering harmonic forms a residual type a posteriori error estimate is obtained by using the hodge decomposition the regular decomposition and bounded commuting quasi interpolants an additional marking strategy is added to ensure the quasi orthogonality based on which the convergence of adaptive mixed finite element methods is obtained without assuming the initial mesh size is small enough

schwarz methods of neumann neumann type for three dimensional elliptic finite element problems several domain decomposition methods of neumann neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements we will only consider problems in three dimensions several new variants of the basic algorithm are introduced in a schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi level methods the neumann neumann algorithms have several advantages over other domain decomposition methods the subregions which define the subproblems only share the boundary degrees of freedom with their neighbors the subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries in its original form however the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases in the new variants of the algorithms considered in this paper the preconditioners include global components of low rank to overcome this difficulty bounds are established for the condition number of the iteration operator which are independent of the number of subregions and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions 1995 john wiley sons inc

on generalizing the algebraic multigrid framework we present a theory for algebraic multigrid amg methods that allows for general smoothing processes and general coarsening approaches the goal of the theory is to provide guidance in the development of new more robust amg algorithms in particular we introduce several compatible relaxation methods and give theoretical justification for their use as tools for measuring the quality of coarse grids

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

convergence and optimality of adaptive edge finite element methods for time harmonic maxwell equations we consider a standard adaptive edge finite element method aefem based on arbitrary order nedelec edge elements for three dimensional indefinite time harmonic maxwell equations we prove that the aefem gives a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops provided the initial mesh is fine enough using the geometric decay we show that the aefem yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom the main technical contribution of the paper is in the establishment of a quasi orthogonality and a localized a posteriori error estimator

fourier series expansion in a non orthogonal system of coordinates for the simulation of 3d dc borehole resistivity measurements abstract we describe a new method to simulate 3d borehole resistivity measurements at zero frequency dc the method combines the use of a fourier series expansion in a non orthogonal system of coordinates with an existing 2d goal oriented higher order self adaptive hp finite element algorithm the new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells it delivers high accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3d simulators since the number of fourier modes basis functions needed to solve practical applications is limited typically below 10 furthermore numerical results indicate that the optimal 2d grid based on the 0th fourier mode also called central fourier mode can be employed to efficiently solve the final 3d problem thereby avoiding the expensive construction of optimal 3d grids specifically for a challenging through casing resistivity application we reduce the computational time from several days using a 3d simulator to just 2 h with the new method while gaining accuracy the new simulation method can be easily extended to different physical phenomena with similar geometries as those arising in the simulation of 3d borehole electrodynamics and sonic acoustics coupled with elasticity measurements in addition the method is especially suited for inversion since we demonstrate that the number of fourier modes needed for the exact representation of the materials is limited to only one the central mode for the case of borehole measurements acquired in deviated wells

an improved algebraic multigrid method for solving maxwell s equations we propose two improvements to the reitzinger and schoberl algebraic multigrid amg method for solving the eddy current approximations to maxwell s equations the main focus in the reitzinger schoberl method is to maintain null space properties of the weak nabla times nabla times operator on coarse grids while these null space properties are critical they are not enough to guarantee h independent convergence of the overall multigrid method we illustrate how the reitzinger schoberl amg method loses h independence due to the somewhat limited approximation property of the grid transfer operators we present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates the first improvement is based on smoothing the reitzinger schoberl grid transfer operators the second improvement is obtained by using higher order nodal interpolation to derive the corresponding amg interpolation operators while not completely h independent the resulting amg cg method demonstrates improved convergence behavior while maintaining low operator complexity

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

analysis of a multigrid algorithm for time harmonic maxwell equations this paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic maxwell equations in particular a backslash multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used the method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator this idea has previously been used in multigrid analysis of indefinite second order elliptic problems however the maxwell application involves a nonelliptic indefinite operator with the help of a few new estimates the earlier ideas can still be applied some numerical experiments with lowest order nedelec elements are also reported

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

integration of hp adaptivity and a two grid solver for electromagnetic problems we present implementation details and analyze convergence of a two grid solver forming the core of a fully automatic hp adaptive strategy for electromagnetic problems the solver delivers a solution for a fine grid obtained from an arbitrary coarse hp grid by a global hp refinement the classical v cycle algorithm combines an overlapping block jacobi smoother with optimal relaxation and a direct solve on the coarse grid a theoretical analysis of the two grid solver is illustrated with numerical experiments several electromagnetic applications show the efficiency of combining the fully automatic hp adaptive strategy with the two grid solver

exact de rham sequences of spaces defined on macro elements in two and three spatial dimensions this paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure the main requirement is that the agglomerates share faces that have closed boundaries composed of 1 d edges the spaces resulting from the agglomerated elements are subspaces of the original de rham sequence of h 1 conforming h mathbf curl conforming h mathbf div conforming and piecewise constant spaces associated with an unstructured fine mesh the procedure can be recursively applied so that a sequence of nested de rham complexes can be constructed as an illustration we generate coarser spaces from the sequence corresponding to the lowest order nedelec spaces lowest order raviart thomas spaces and for piecewise linear h 1 conforming spaces all in three dimensions the resulting v cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case

multigrid in h div on axisymmetric domains abstract in this paper we will construct and analyze a multigrid algorithm that can be applied to weighted h div problems on a two dimensional domain these problems arise after performing a dimension reduction to a three dimensional axisymmetric h div problem we will use recently developed fourier finite element spaces that can be applied to axisymmetric h div problems with general data we prove that if the axisymmetric domain is convex then the multigrid v cycle with modern smoothers will converge uniformly with respect to the meshsize

a feti domain decomposition method for edge element approximations in two dimensions with discontinuous coefficients a class of finite element tearing and interconnecting feti methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed first an abstract framework is presented for the analysis of a class of feti methods where a natural coarse problem associated with the substructures is lacking then a family of feti methods for edge element approximations is proposed it is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures the estimate is also independent of the jumps of both of the coefficients of the original problem numerical results validating our theoretical bounds are given the method and its analysis can be easily generalized to raviart thomas element approximations in two and three dimensions

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens

an algebraic multigrid approach based on a compatible gauge reformulation of maxwell s equations with the rise in popularity of compatible finite element finite difference and finite volume discretizations for the time domain eddy current equations there has been a corresponding need for fast solvers of the resulting linear algebraic systems however the traits that make compatible discretizations a preferred choice for the maxwell s equations also render these linear systems essentially intractable by truly black box techniques we propose an algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid amg technique for this reformulated problem the reformulation process takes advantage of a discrete hodge decomposition on cochains to replace the discrete eddy current equations by an equivalent 2 times2 block linear system whose diagonal blocks are discrete hodge laplace operators acting on 1 cochains and 0 cochains respectively while this new amg technique requires somewhat specialized treatment on the finest mesh the coarser meshes can be handled using standard methods for laplace type problems our new amg method is applicable to a wide range of compatible methods on structured and unstructured grids including edge finite elements mimetic finite differences covolume methods and yee like schemes we illustrate the new technique using edge elements in the context of smoothed aggregation amg and present computational results for problems in both two and three dimensions