a staggered discontinuous galerkin method for wave propagation in media with dielectrics and meta materials some electromagnetic materials exhibit in a given frequency range effective dielectric permittivity and or magnetic permeability which are negative in the literature they are called negative index materials left handed materials or meta materials we propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta material the method we investigate can be considered as an alternative method compared to the method presented by the second author and co workers in particular we shall use the abstract framework they developed to prove well posedness of the exact problem we recast this problem to fit later discretization by the staggered discontinuous galerkin method developed by the first author and co worker a method which relies on introducing an auxiliary unknown convergence of the numerical method is proven with the help of explicit inf sup operators and numerical examples are provided to show the efficiency of the method

the ngp stability of runge kutta methods for systems of neutral delay differential equations this paper deals with the stability analysis of implicit runge kutta methods for the numerical solutions of the systems of neutral delay differential equations we focus on the behavior of such methods with respect to the linear test equations where tau 0 l m and n are d times d complex matrices we show that an implicit runge kutta method is ngp stable if and only if it is a stable

constraint energy minimizing generalized multiscale discontinuous galerkin method abstract numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas however numerical solutions on the fine grid are often prohibitively expensive and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid in this paper we propose an energy minimization based multiscale model reduction approach in the discontinuous galerkin discretization setting the main idea of the method is to extract the non decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems the multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous galerkin formulation the minimal degree of freedom in representing high contrast features is achieved through the design of local spectral problems which provides the most compressed local multiscale space we analyze the method for solving darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast provided that the oversampling size is appropriately chosen numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high contrast heterogeneous media

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games we introduce a near linear complexity geometric and meshless algebraic multigrid multiresolution method for pdes with rough l infty coefficients with rigorous a priori accuracy and performance estimates the method is discovered through a decision game theory formulation of the problems of 1 identifying restriction and interpolation operators 2 recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator and 3 gambling on the value of the solution of the pde based on a hierarchy of nested measurements of its solution or source term the resulting elementary gambles form a hierarchy of deterministic basis functions of h 1 0 omega gamblets that 1 are orthogonal across subscales subbands with respect to the scalar product induced by the energy norm of the pde 2 enable sparse compression of the solution space in h 1 0 omega and 3 induce an orthogonal multiresolution operator decomposition the operating diagram of the multig

a spectral multiscale hybridizable discontinuous galerkin method for second order elliptic problems abstract we design a multiscale model reduction framework within the hybridizable discontinuous galerkin finite element method our approach uses local snapshot spaces and local spectral decomposition following the concept of generalized multiscale finite element methods we propose several multiscale finite element spaces on the coarse edges that provide a reduced dimensional approximation for numerical traces within the hdg framework we provide a general framework for systematic construction of multiscale trace spaces using local snapshots we avoid high dimensional representation of trace spaces and use some local features of the solution space in constructing a low dimensional trace space we investigate the solvability and numerically study the performance of the proposed method on a representative number of numerical examples

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

dynamic data driven bayesian gmsfem abstract in this paper we propose a bayesian approach for multiscale problems with the availability of dynamic observational data our method selects important degrees of freedom probabilistically in a generalized multiscale finite element method framework due to scale disparity in many multiscale applications computational models cannot resolve all scales dominant modes in the generalized multiscale finite element method are used as permanent basis functions which we use to compute an inexpensive multiscale solution and the associated uncertainties through our bayesian framework we can model approximate solutions by selecting the unresolved scales probabilistically we consider parabolic equations in heterogeneous media the temporal domain is partitioned into subintervals using residual information and given dynamic data we design appropriate prior distribution for modeling missing subgrid information the likelihood is designed to minimize the residual in the underlying pde problem and the mismatch of observational data using the resultant posterior distribution the sampling process identifies important degrees of freedom beyond permanent basis functions the method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations

nonlinear stability of runge kutta methods for neutral delay differential equations this paper deals with stability properties of runge kutta methods for the initial value problem in nonlinear neutral delay differential equationsy t f t y t y t t y t t t 0 y t f t t t 0 the new concepts of gs l stability gas l stability and weak gas l stability are introduced and it is shown that k l algebraically stable runge kutta methods with piecewise linear interpolation are gs l gas l and weakly gas l stable two numerical examples are given in the end of this paper which confirm our results

generalized multiscale multicontinuum model for fractured vuggy carbonate reservoirs abstract simulating flow in a highly heterogeneous reservoir with multiscale characteristics is computationally demanding to tackle this problem we propose a numerical scheme based on the generalized multiscale finite element method gmsfem and a triple continuum model the aim is to obtain a more efficient numerical approach that can explicitly represent the interactions among different continua to enhance the applicability of our proposed model we combine the discrete fracture model dfm to model the local effects of discrete fractures in the proposed model the gmsfem as an advanced model reduction technique enables capturing the multiscale flow dynamics this is accomplished by systematically generating an approximation space through solving a series of local snapshot and spectral problems the resulting eigenfunctions can pass the local features to the global level when acting as basis functions in the global coarse problems our goal in this paper is to further improve the accuracy of flow simulation in complicated reservoirs especially for the case when multiple discrete fractures located in single coarse neighborhood several numerical experiments are conducted to confirm the success of our proposed method and a rigorous convergence proof is also given

analysis of hdg methods for stokes flow in this paper we analyze a hybridizable discontinuous galerkin method for numerically solving the stokes equations the method uses polynomials of degree k for all the components of the approximate solution of the gradient velocity pressure formulation the novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method it renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges as a consequence we prove that the approximations of the velocity gradient the velocity and the pressure converge with the optimal order of convergence of k 1 in l 2 for any k 0 moreover taking advantage of the superconvergence properties of the velocity we introduce a new element by element postprocessing to obtain a new velocity approximation which is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 numerical experiments are presented which validate the theoretical results

multiscale finite element methods for high contrast problems using local spectral basis functions in this paper we study multiscale finite element methods msfems using spectral multiscale basis functions that are designed for high contrast problems multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem this local spectral problem strongly depends on the choice of initial partition of unity functions the resulting space enriches the initial multiscale space using eigenvectors of local spectral problem the eigenvectors corresponding to small asymptotically vanishing eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small asymptotically vanishing eigenvalues we present a convergence study that shows that the convergence rate in energy norm is proportional to h 1 2 where is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space thus we would like to reach to a larger eigenvalue with a smaller coarse space this is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems numerical results are presented to back up our theoretical results and to show higher accuracy of msfems with spectral multiscale basis functions we also present a hierarchical construction of the eigenvectors that provides cpu savings

series with general exponents abstract we define the artinian and noetherian algebra which consist of formal series involving exponents which are not necessarily integers all of the usual operations are defined here and characterized as an application we compute the algebra of symmetric functions with nonnegative real exponents the applications to logarithmic series and the umbral calculus are deferred to another paper 9

staggered discontinuous galerkin methods for the incompressible navier stokes equations in this paper we present a staggered discontinuous galerkin method for the approximation of the incompressible navier stokes equations our new method combines the advantages of discontinuous galerkin methods and staggered meshes and results in many good properties namely local and global conservations optimal convergence and superconvergence through the use of a local postprocessing technique another key feature is that our method provides a skew symmetric discretization of the convection term with the aim of giving a better conservation property compared with existing discretizations we will present extensive numerical results including kovasznay flow taylor vortex flow lid driven cavity flow parallel plate flow and channel expansion flow to show the performance of the method

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

variational multiscale analysis the fine scale green s function projection optimization localization and stabilized methods we derive an explicit formula for the fine scale green s function arising in variational multiscale analysis the formula is expressed in terms of the classical green s function and a projector which defines the decomposition of the solution into coarse and fine scales the theory is presented in an abstract operator format and subsequently specialized for the advection diffusion equation it is shown that different projectors lead to fine scale green s functions with very different properties for example in the advection dominated case the projector induced by the h 1 0 seminorm produces a fine scale green s function which is highly attenuated and localized these are very desirable properties in a multiscale method and ones that are not shared by the l 2 projector by design the coarse scale solution attains optimality in the norm associated with the projector this property combined with a localized fine scale green s function indicates the possibility of effective methods with local character for dominantly hyperbolic problems the constructs lead to a new class of stabilized methods and the relationship between h 1 0 optimality and the streamline upwind petrov galerkin supg method is described

the iterated logarithmic algebra ii sheffer sequences abstract an extension of the theory of the iterated logarithmic algebra 1 gives the logarithmic analog of a sheffer or appell sequence of polynomials this leads to several examples including stirling s formula and a logarithmic version of the euler maclaurin summation formula

convergence of a discontinuous galerkin multiscale method we present a discontinuous galerkin multiscale method for second order elliptic problems and prove convergence we consider a heterogeneous and highly varying diffusion coefficient in l infty omega mathbb r d times d sym with uniform spectral bounds without any assumption on scale separation or periodicity the multiscale method uses a corrected basis that is computed on patches subdomains the error due to truncation of the corrected basis decreases exponentially with the size of the patches hence to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size h to a reference solution it is sufficient to choose the patch sizes mathcal o h log h we also discuss a way to further localize the corrected basis to elementwise support improved convergence rate can be achieved depending on the piecewise regularity of the forcing function linear convergence in energy norm and quadratic convergence in the l 2 norm is obtained independently of the

analysis of a multiscale discontinuous galerkin method for convection diffusion problems we study a multiscale discontinuous galerkin method introduced in t j r hughes g scovazzi p bochev and a buffa comput meth appl mech engrg 195 2006 pp 2761 2787 that reduces the computational complexity of the discontinuous galerkin method seemingly without adversely affecting the quality of results for a stabilized variant we are able to obtain the same error estimates for the convection diffusion equation as for the usual discontinuous galerkin method we assess the stability of the unstabilized case numerically and find that the inf sup constant is positive bounded uniformly away from zero and very similar to that for the usual discontinuous galerkin method

nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media abstract our goal of this paper is to develop a new upscaling method for multicontinua flow problems in fractured porous media we consider a system of equations that describes flow phenomena with multiple flow variables defined on both matrix and fractures to construct our upscaled model we will apply the nonlocal multicontinua nlmc upscaling technique the upscaled coefficients are obtained by using some multiscale basis functions which are solutions of local problems defined on oversampled regions for each continuum within a target coarse element we will solve a local problem defined on an oversampling region obtained by extending the target element by few coarse grid layers with a set of constraints which enforce the local solution to have mean value one on the chosen continuum and zero mean otherwise the resulting multiscale basis functions have been shown to have good approximation properties to illustrate the idea of our approach we will consider a dual continua background model consisting of discrete fractures in two space dimensions that is we consider a system with three continua we will present several numerical examples and they show that our method is able to capture the interaction between matrix continua and discrete fractures on the coarse grid efficiently

generalized multiscale finite element method symmetric interior penalty coupling motivated by applications to numerical simulations of flows in highly heterogeneous porous media we develop multiscale finite element methods for second order elliptic equations we discuss a multiscale model reduction technique in the framework of the discontinuous galerkin finite element method we propose two different finite element spaces on the coarse mesh the first space is based on a local eigenvalue problem that uses an interior weighted l 2 norm and a boundary weighted l 2 norm for computing the mass matrix the second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension the approximation with these multiscale spaces is based on the discontinuous galerkin finite element method framework we investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

the interated logarithmic algebra abstract we generalize the umbral calculus of g c rota adv in math 27 1978 95 188 by studying not only sequences of polynomials and inverse power series or even the logarithms studied by d loeb and g c rota adv in math 75 1989 1 118 but instead we study sequences of formal expressions involving the iterated logarithms and x to an arbitrary real power using a theory of formal power series with real exponents and a more general definition of factorial binomial coefficient and stirling numbers to all the real numbers we define the iterated logarithmic algebra j its elements are the formal representations of the asymptotic expansions of a large class of real functions and we define the harmonic logarithm basis of j which will be interpreted as a generalization of the powers xn since it behaves nicely with respect to the derivative we classify all operators over j which commute with the derivative classically there are known as shift invariant operators and formulate several equivalent definitions of a sequence of binomial type we then derive many formulas useful towards the calculation of these sequences including the recurrence formula the transfer formula and the lagrange inversion formula in the sequel 22 we study sheffer sequences and give many examples

residual driven online generalized multiscale finite element methods the construction of local reduced order models via multiscale basis functions has been an area of active research in this paper we propose online multiscale basis functions which are constructed using the offline space and the current residual online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators we derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error this error decrease is independent of physical parameters such as the contrast and multiple scales in the problem the offline spaces are constructed using generalized multiscale finite element methods gmsfem we show that if one chooses a sufficient number of offline basis functions one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast we note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects using the residual information we can incorporate the distant information provided the offline approximation satisfies certain properties in the paper theoretical and numerical results are presented our numerical results show that if the offline space is sufficiently large in terms of the dimension such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues then the error reduction by adding online multiscale basis function is independent of the contrast we discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed