a multiscale direct solver for the approximation of flows in high contrast porous media abstract we consider a non overlapping domain decomposition approach to approximate the solution of elliptic boundary value problems with high contrast in their coefficients we propose a method such that initially local solutions subject to robin boundary conditions in each primal subdomain are constructed with locally conservative finite element or finite volume methods then a novel approach is introduced to obtain a discontinuous global solution in terms of linear combination of the local subdomain solutions in the proposed algorithm the computation of local solutions for unions of subdomains are localized at nearest neighbor subdomain boundaries thus avoiding the solution of global interface problems we remove discontinuities in a smoothing step that is defined on a staggered grid or dual subdomains the resulting algorithm is naturally parallelizable and can be employed as a parallel direct solver offering great potential for the numerical solution of large problems in fact subdomains can be considered small enough to fit well in gpus and the proposed procedure can handle adaptive in space simulations effectively numerical simulations are presented and discussed we demonstrate the effectiveness of the proposed approach with two and three dimensional high contrast and channelized coefficients that lead to challenging approximation problems the new procedure although designed for parallel processing is also of value for serial calculations

a least squares based weak galerkin finite element method for second order elliptic equations in this article we introduce a least squares based weak galerkin finite element method for the second order elliptic equation this new method is shown to provide very accurate numerical approximations for both the primal and the flux variables in contrast to other existing least squares finite element methods this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon polyhedron shapes we also develop a schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element optimal order error estimates for both the primal and the flux variables are established an extensive set of numerical experiments are conducted to demonstrate the robustness reliability flexibility and accuracy of the least squares based weak galerkin finite element method the numerical examples cover a wide range of applied problems including singularly

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

subgrid upscaling and mixed multiscale finite elements second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale epsilon can be approximated on coarse meshes of spacing h gg epsilon only if one uses special techniques the mixed variational multiscale method also called subgrid upscaling can be used and this method is extended to allow oversampling of the local subgrid problems the method is shown to be equivalent to the multiscale finite element method when one uses the lowest order raviart thomas spaces and provided that there are no fine scale components in the source function f in the periodic setting a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is o epsilon hm sqrt epsilon h where m 1 moreover m 2 if one uses the second order brezzi douglas marini or brezzi douglas duran fortin spaces and no oversampling the error bounding constant depends only on the hm 1 norm of f and so is independent of small scales when m 1 when oversampling is not used a superconvergence result for the pressure approximation is shown

numerical multiscale methods for a reaction dominated model a galerkin enriched finite element method gem is proposed for the singularly perturbed reaction diffusion equation this new method is an improvement on the petrov galerkin enriched method pgem where now the standard piecewise bi linear test space incorporates fine scales this appears as the fundamental ingredient for suppressing oscillations in the numerical solutions also new parameter free stabilized finite element methods derived from both the gem and the pgem are driven by local generalized eigenvalue problems in the process jump stabilizing terms belonging to the class of cip methods emerge as a result of the enriching procedure interestingly numerical results indicate that jump based stabilizations are unnecessary and sometimes undesirable when treating reaction dominated problems finally we establish relationships with more standard enriched and stabilized methods and show that the proposed methods outperform them numerically

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

superconvergence in a dpg method for an ultra weak formulation abstract in this work we study a dpg method for an ultra weak variational formulation of a reaction diffusion problem we improve existing a priori convergence results by sharpening an approximation result for the numerical flux by duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one furthermore we introduce a simple elementwise postprocessing of the solution and prove superconvergence numerical experiments indicate that the obtained results are valid beyond the underlying model problem

first order least squares method with weakly imposed boundary condition for convection dominated diffusion problems we present and analyze a first order least squares method for convection dominated diffusion problems which provides robust l 2 a priori error estimate for the scalar variable even if the given data f l 2 the novel theoretical approach is to rewrite the method in the framework of discontinuous petrov galerkin dpg method and then show numerical stability by using a key equation discovered by gopalakrishnan and qiu 2014 this new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations we also show that the condition number of the global matrix is independent of the diffusion coefficient a key feature of the method is that there is no stabilization parameter chosen empirically in addition dirichlet boundary condition is weakly imposed numerical experiments verify our theoretical results and in particular show our way of weakly imposing dirichlet boundary condition is essential to the design of least squares methods numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes which are unstructured quasi uniform

multiscale hybrid mixed method for the stokes and brinkman equations the method abstract the multiscale hybrid mixed mhm method is extended to the stokes and brinkman equations with highly heterogeneous coefficients the approach is constructive we first propose an equivalent dual hybrid formulation of the original problem using a coarse partition of the heterogeneous domain faces may be not aligned with jumps in the data then the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent stokes or brinkman problems at the continuous level owing to this decomposition the one level mhm method stems from the standard galerkin approach for the lagrange multiplier space basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation they are the exact solution of the local problems with prescribed neumann boundary conditions on faces driven by the lagrange multipliers we make the mhm method effective by adopting the unusual stabilized finite element method to solve the local problems approximately as such equal order interpolation turns out to be an option for the velocity the pressure and the lagrange multipliers the numerical solutions share the important properties of the continuum such as local equilibrium with respect to external forces and local mass conservation several academic and highly heterogeneous tests infer that the method achieves super convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms

the multiscale hybrid mixed method in general polygonal meshes this work extends the general form of the multiscale hybrid mixed mhm method for the second order laplace darcy equation to general non conforming polygonal meshes the main properties of the mhm method i e stability optimal convergence and local conservation are proven independently of the geometry of the elements used for the first level mesh more precisely it is proven that piecewise polynomials of degree k and k 1 k ge 0 for the lagrange multipliers flux along with continuous piecewise polynomial interpolations of degree k 1 posed on second level sub meshes are stable if the latter is fine enough with respect to the mesh for the lagrange multiplier we provide an explicit sufficient condition for this restriction also we prove that the error converges with order k 1 and k 2 in the broken h 1 and l 2 norms respectively under usual regularity assumptions and that such estimates also hold for non convex or even non simply connected elements numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides

the multiscale robin coupled method for flows in porous media abstract a multiscale mixed method aiming at the accurate approximation of velocity and pressure fields in heterogeneous porous media is proposed the procedure is based on a new domain decomposition method in which the local problems are subject to robin boundary conditions the domain decomposition procedure is defined in terms of two independent spaces on the skeleton of the decomposition corresponding to interface pressures and fluxes that can be chosen with great flexibility to accommodate local features of the underlying permeability fields the well posedness of the new domain decomposition procedure is established and its connection with the method of douglas et al 1993 12 is identified also allowing us to reinterpret the known procedure as an optimized schwarz or two lagrange multiplier method the multiscale property of the new domain decomposition method is indicated and its relation with the multiscale mortar mixed finite element method mmmfem and the multiscale hybrid mixed mhm finite element method is discussed numerical simulations are presented aiming at illustrating several features of the new method initially we illustrate the possibility of switching from mmmfem to mhm by suitably varying the robin condition parameter in the new multiscale method then we turn our attention to realistic flows in high contrast channelized porous formations we show that for a range of values of the robin condition parameter our method provides better approximations for pressure and velocity than those computed with either the mmmfem and the mhm this is an indication that our method has the potential to produce more accurate velocity fields in the presence of rough realistic permeability fields of petroleum reservoirs

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

the dpg star method abstract this article introduces the dpg star from now on denoted dpg finite element method it is a method that is in some sense dual to the discontinuous petrov galerkin dpg method the dpg methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem in the same vein the dpg methodology is a means to solve an underdetermined discretization these two viewpoints are developed by embedding the same operator equation into two different saddle point problems the analyses of the two problems have many common elements comparison to other methods in the literature round out the newly garnered perspective notably dpg and dpg methods can be seen as generalizations of l l and least squares methods respectively a priori error analysis and a posteriori error control for the dpg method are considered in detail reports of several numerical experiments are provided which demonstrate the essential features of the new method a notable difference between the results from the dpg and dpg analyses is that the convergence rates of the former are limited by the regularity of an extraneous lagrange multiplier variable

homology of the zero set of a nilpotent vector field on a flag manifold 0 1 let x be a linear transformation of a finite dimensional vector space v the configuration of flags in v which are fixed by x has rather remarkable properties when x is unipotent though this case is especially interesting the proper generality in which to study such configurations is in the theory of reductive algebraic groups where their definition can be reformulated in the language of borel subalgebras as follows let g be a connected reductive group over c with lie algebra g and let n e g be a nilpotent element let q be the variety of all borel subalgebras of g and let

a petrov galerkin multilayer discretization to second order elliptic boundary value problems abstract we study in this paper a multilayer discretization of second order elliptic problems aimed at providing reliable multilayer discretizations of shallow fluid flow problems with diffusive effects this discretization is based upon the formulation by transposition of the equations it is a petrov galerkin discretization in which the trial functions are piecewise constant per horizontal layers while the test functions are continuous piecewise linear on a vertically shifted grid we prove the well posedness and optimal error order estimates for this discretization in natural norms based upon specific inf sup conditions we present some numerical tests with parallel computing of the solution based upon the multilayer structure of the discretization for academic problems with smooth solutions with results in full agreement with the theory developed

a dual petrovgalerkin finite element method for the convectiondiffusion equation we present a minimum residual finite element method based on a dual petrovgalerkin formulation for convectiondiffusion problems in a higher order adaptive continuous galerkin setting the method borrows concepts from both the discontinuous petrovgalerkin dpg method by demkowicz and gopalakrishnan 2011 and the method of variational stabilization by cohen dahmen and welper 2012 and it can also be interpreted as a variational multiscale method in which the fine scales are defined through a dual orthogonality condition a key ingredient in the method is the proper choice of dual norm used to measure the residual and we present two choices which are observed to be robust in both convection and diffusion dominated regimes as well as a proof of stability for quasi uniform meshes and a method for the weak imposition of boundary conditions numerically obtained convergence rates in 2d are reported and benchmark numerical examples are given to illustrate the behavior of the method

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

low order dpg fem for an elliptic pde this paper introduces a novel lowest order discontinuous petrovgalerkin dpg finite element method fem for the poisson model problem the ultra weak formulation allows for piecewise constant and affine ansatz functions and for piecewise affine and lowest order raviartthomas test functions this lowest order discretization for the poisson model problem allows for a direct proof of the discrete infsup condition and a complete apriori and aposteriori error analysis numerical experiments investigate the performance of the method and underline the quasi optimal convergence

a discontinuous petrov galerkin method with lagrangian multipliers for second order elliptic problems we present a discontinuous petrov galerkin dpg method for the finite element discretization of second order elliptic boundary value problems the novel approach emanates from a one element weak formulation of the differential problem this procedure which is typical of discontinuous galerkin dg methods is based on introducing variables defined in the interior and on the boundary of the element the interface variables are suitable lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative thus providing the proper connection between neighboring elements the internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole lagrangian multipliers a stability and convergence analysis of the novel formulation is carried out and its connection with mixed hybrid and dg methods is explored numerical tests on several benchmark problems are included to validate the convergence performance and the flux conservation properties of the dpg method

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

one dimensional wave propagation in a functionally graded elastic medium abstract in this study the one dimensional wave propagation in a functionally graded elastic slab is considered it is assumed that the stiffness and density of the medium vary continuously in thickness direction and it is initially at rest and stress free the slab is subjected to a pressure pulse on one surface and a vanishing stress or displacement condition on the other the solution is obtained in wave summation form propagation of a rectangular pressure pulse in a graded medium that consists of either nickel zirconia or aluminium silicon carbide is studied as examples it is shown that there is considerable wave distortion in time and the distortion is much more pronounced in slabs with fixed free boundary conditions a simple approximate expression giving the peak stress is developed also it is demonstrated that the energy balance principle may be used as a convergence criterion in the calculation of stresses

a petrov galerkin enriched method a mass conservative finite element method for the darcy equation starting from the non stable p1 p0 discretization we build enhanced methods for the darcy equation which are stable and locally mass conservative the methods are derived in a petrov galerkin framework where both velocity and pressure trial spaces are enriched with multiscale functions these functions solve local problems correcting the residuals of the strong equations in each element and interior edge which leads to a velocity space enhanced with functions belonging to the lowest order raviart thomas space several numerical tests validate the methods

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

a pde constrained optimization approach to the discontinuous petrov galerkin method with a trust region inexact newton cg solver abstract we propose a partial differential equation constrained pde constrained approach to the discontinuous petrov galerkin dpg of demkowicz and gopalakrishnan 2010 2011 this view opens the door to invite all the state of the art pde constrained techniques to be part of the dpg framework and hence enabling one to solve large scale and difficult nonlinear problems efficiently that is our proposed method preserves all the attractive features of the dpg framework while enjoying all advances from the pde constrained optimization community the proposed approach can be considered as a rayleigh ritz method for the dpg minimum residual statement it is equipped with a trust region inexact newton conjugate gradient trincg method which prevents over solving when optimization iterates are far away from the optimal solution but converges quadratically otherwise for sufficiently smooth residual the pde constrained approach together with the trincg solver is therefore a robust iterative method for the dpg framework it is robust in the sense that the approximate solution is improved after each optimization iterate until the algorithm converges to a local minimum of the residual as numerically shown it is also scalable in the sense that the number of newton iterations remains constant as either the mesh or the solution order is refined moreover the proposed approach solves neither the optimal test functions nor the original pde directly though the discretization of the latter is still necessary the gradient and hessian vector product which are required by the trincg solver are explicitly derived for both abstract variational problems and viscous burger equation this reveals a fact that the complexity of each newton iteration scales like o n n c g where n and n c g are the number of unknowns and the number of conjugate iterations respectively optimal h and p convergences of the proposed approach are demonstrated for laplace and helmholtz equations numerical results for the viscous burger equation and the euler equation of gas dynamics are very promising

robust coupling of dpg and bem for a singularly perturbed transmission problem abstract we consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the laplacian in the exterior connected through standard transmission conditions we establish a dpg scheme coupled with galerkin boundary elements for its discretization and prove its robustness for the field variables in so called balanced norms our coupling scheme is the one from fuhrer et al 2016 adapted to the singularly perturbed case by using the scheme from heuer and karkulik 2015 essential feature of our method is that optimal test functions have to be computed only locally we report on various numerical experiments in two dimensions

a quadratic layer element for analyzing stress waves in fgms and its application in material characterization abstract a novel method is presented for investigating elastic waves in functionally graded material fgm plates excited by plane pressure waves the fgm plate is first divided into quadratic layer elements qles a general solution for the equation of motion governing the qle has been derived the general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the frequency domain for an arbitrary fgm plate the response of the plate to an incident pressure wave is obtained using the fourier transform techniques results obtained by the present method are compared with an existing method using homogeneous layer elements numerical examples are presented to investigate stress waves in fgm plates the relationship between the surface displacement response and the material property of quadratic fgm plates has been analytically obtained for the material characterization a computational inverse technique is also presented for characterizing material property of an arbitrary fgm plate from the surface displacement response data using present qle method as forward solver and genetic algorithm as the inverse operator this technique is utilized to reconstruct the material property of an actual sic c fgm

the dpg method for the stokes problem we discuss well posedness and convergence theory for the dpg method applied to a general system of linear partial differential equations pdes and specialize the results to the classical stokes problem the stokes problem is an iconic troublemaker for standard bubnov galerkin methods if discretizations are not carefully designed they may exhibit non convergence or locking by contrast dpg does not require us to treat the stokes problem in any special manner we illustrate and confirm our theoretical convergence estimates with numerical experiments

a robust dpg method for convection dominated diffusion problems ii adjoint boundary conditions and mesh dependent test norms we introduce a dpg method for convection dominated diffusion problems the choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter demkowicz and heuer 2011 18 we propose a new inflow boundary condition which regularizes the adjoint problem allowing the use of a stronger test norm the robustness of the method is proven and numerical experiments demonstrate the method s robust behavior

a multiscale hybrid method for darcy s problems using mixed finite element local solvers abstract multiscale hybrid mixed mhm method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions for fluid flow normal fluxes multiplier over macro element boundaries and coarse piecewise constant potential approximations in each macro element are computed upscaling then small details are resolved by local problems using fine representations inside the macro elements setting the multiplier as neumann boundary conditions downscaling in this work a variant of the method is developed denoted by mhm h div adopting mixed finite elements at the downscaling stage instead of continuous finite elements used in all previous publications of the method thus this alternative mhm method inherits improvements typical of mixed methods as better flux accuracy and local mass conservation at the micro scale level inside the macro elements which are important properties for multi phase flows in rough heterogeneous media different two scale stable space settings are considered vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries in each macro element the internal flux components with vanishing normal traces and the potential approximations may be enriched in different extents with respect to internal mesh size internal polynomial degree or both the choice being determined by the problem at hands a unified general error analysis of the mhm h div method is presented for all these two scale space scenarios both mhm versions are compared for 2d test problems with smooth solutions for convergence rates verification and for darcy s flow in heterogeneous media mhm h div 3d simulations are presented for a known singular darcy s solution using adaptive macro partitions and for an oscillatory permeability scenario

locally conservative discontinuous petrov galerkin finite elements for fluid problems we develop a locally conservative formulation of the discontinuous petrov galerkin finite element method dpg for convection diffusion type problems using lagrange multipliers to exactly enforce conservation over each element we provide a proof of convergence as well as extensive numerical experiments showing that the method is indeed locally conservative we also show that standard dpg while not guaranteed to be conservative is nearly conservative for many of the benchmarks considered the new method preserves many of the attractive features of dpg but turns the normally symmetric positive definite dpg system into a saddle point problem

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

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

the dpg methodology applied to different variational formulations of linear elasticity abstract the flexibility of the dpg methodology is exposed by solving the linear elasticity equations under different variational formulations including some with non symmetric functional settings different infinite dimensional trial and test spaces the family of formulations presented is proved to be mutually ill or well posed when using traditional energy spaces on the whole domain moreover they are shown to remain well posed when using broken energy spaces and interface variables four variational formulations are solved in 3d using the dpg methodology numerical evidence is given for both smooth and singular solutions and the expected convergence rates are observed

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

a unified discontinuous petrov galerkin method and its analysis for friedrichs systems we propose a unified discontinuous petrov galerkin dpg framework with optimal test functions for friedrichs like systems which embrace a large class of elliptic parabolic and hyperbolic partial differential equations pdes the well posedness i e existence uniqueness and stability of the dpg solution is established on a single abstract dpg formulation and two abstract dpg methods corresponding to two different but equivalent norms are devised we then apply the single dpg framework to several linear ized pdes including but not limited to scalar transport laplace diffusion convection diffusion convection diffusion reaction linear ized continuum mechanics e g linear ized elasticity a version of linearized navier stokes equations etc time domain acoustics and a version of the maxwell s equations the results show that we not only recover several existing dpg methods but also discover new dpg methods for both pdes currently considered in the dpg community and new ones as

a two level enriched finite element method for a mixed problem the simplest pair of spaces is made inf sup stable for the mixed form of the darcy equation the key ingredient is to enhance the finite element spaces inside a petrov galerkin framework with functions satisfying element wise local darcy problems with right hand sides depending on the residuals over elements and edges the enriched method is symmetric locally mass conservative and keeps the degrees of freedom of the original interpolation spaces first we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms then a low cost two level finite element method is proposed to effectively obtain enhancing basis functions the approach lays on a two scale numerical analysis and shows that well posedness and optimality is kept despite the second level numerical approximation several numerical experiments validate the theoretical results and compares favourably in some cases our results with the classical raviart thomas element

polynomial extension operators part ii consider the tangential trace of a vector polynomial on the surface of a tetrahedron we construct an extension operator that extends such a trace function into a polynomial on the tetrahedron this operator can be continuously extended to the trace space of h mathbf curl furthermore it satisfies a commutativity property with an extension operator we constructed in part i of this series such extensions are a fundamental ingredient of high order finite element analysis

a locking free hp dpg method for linear elasticity with symmetric stresses we present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size h and polynomial degree p this is achieved within the recently developed discontinuous petrov galerkin dpg framework in this framework both the stress and the displacement approximations are discontinuous across element interfaces we study locking free convergence properties and the interrelationships between the two dpg methods

towards multiscale functions enriching finite element spaces with local but not bubble like functions in this paper we propose a novel way via finite elements to treat problems that can be singular perturbed a reaction diffusion equation in our case we enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem as in the residual free bubble rfb setting but do not require these functions to vanish on each element edge a departure from the rfb paradigm such multiscale functions have an analytic expression for triangles and rectangles bubbles are the choice for the test functions allowing static condensation thus our method is of petrov galerkin type we perform several numerical validations which confirm the good performance of the method 2004 elsevier b v all rights reserved

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

a low order discontinuous petrov galerkin method for the stokes equations this paper introduces a low order discontinuous petrov galerkin dpg finite element method fem for the stokes equations the ultra weak formulation utilizes piecewise constant and affine ansatz functions and piecewise affine and discontinuous lowest order raviart thomas test search functions this low order discretization for the stokes equations allows for a direct proof of the discrete inf sup condition with explicit constants the general framework of carstensen et al siam j numer anal 52 3 1335 1353 2014 then implies a complete a priori and a posteriori error analysis of the dpg fem in the natural norms numerical experiments investigate the performance of the method and underline its quasi optimal convergence

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

nonlinear discontinuous petrov galerkin methods the discontinuous petrov galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper its lowest order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation a reduced formulation and a weighted nonlinear least squares method quasi optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dpg framework for the approximation of a regular solution the variational model example allows for a built in guaranteed error control despite inexact solve the subtle uniqueness of discrete minimizers is monitored in numerical examples

analysis of the discontinuous petrov galerkin method with optimal test functions for the reissner mindlin plate bending model we analyze the discontinuous petrov galerkin dpg method with optimal test functions when applied to solve the reissner mindlin model of plate bending we prove that the hybrid variational formulation underlying the dpg method is well posed stable with a thickness dependent constant in a norm encompassing the l 2 norms of the bending moment the shear force the transverse deflection and the rotation vector we then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p we show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p 3 we prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non affine mesh sequences

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

local global multiscale model reduction for flows in high contrast heterogeneous media in this paper we study model reduction for multiscale problems in heterogeneous high contrast media our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods see 19 with global model reduction methods such as balanced truncation approaches implemented on a coarse grid local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution in particular our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem global model reduction based on balanced truncation methods is used to identify important global coarse scale modes this provides a substantial cpu savings as lyapunov equations are solved for the coarse system typical local multiscale methods are designed to find an approximation of the solution for any given coarse level inputs in many practical applications a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse level inputs the proposed approaches provide efficient model reduction tools in this direction our numerical results show that only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes one can achieve a balanced and optimal result

wavenumber explicit analysis of a dpg method for the multidimensional helmholtz equation we study the properties of a novel discontinuous petrov galerkin dpg method for acoustic wave propagation the method yields hermitian positive definite matrices and has good pre asymptotic stability properties numerically we find that the method exhibits negligible phase errors otherwise known as pollution errors even in the lowest order case theoretically we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber the mesh size h and the polynomial degree p but the current state of the theory does not fully explain the remarkably good numerical phase errors theoretically comparisons are made with several other recent works that gave wave number explicit estimates numerically comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures the new dpg method is designed following the previously established principles of optimal test functions in addition to the nonstandard test functions in this work we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates

counter propagation of harmonic waves in exponentially graded materials counter propagation and interaction of two ultrasonic harmonic waves in strongly inhomogeneous exponentially graded material is studied deformations of a specimen with two parallel boundaries are described by the five constant nonlinear theory of elasticity one dimensional problems are investigated in detail the influence of material properties variation on the profile of boundary oscillations is clarified the obtained results will be useful in ultrasonic nondestructive material characterization

reflection functors for quiver varieties and weyl group actions we introduce reflectionfunctors on quiver varieties they are hyper kahler isometries between quiver varieties with different parameters related by elements in the weyl group the definition is motivated by the origial reflection functor given by bernstein gelfand ponomarev 1 but they behave much nicely they are isomorphisms and satisfy the weyl group relations as an application we define weyl group representations of homology groups of quiver varieties they are analogues of slodowy s construction of springer representations of the weyl group

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

multiscale hybrid mixed method this work presents a priori and a posteriori error analyses of a new multiscale hybrid mixed method mhm for an elliptic model specially designed to incorporate multiple scales into the construction of basis functions this finite element method relaxes the continuity of the primal variable through the action of lagrange multipliers while assuring the strong continuity of the normal component of the flux dual variable as a result the dual variable which stems from a simple postprocessing of the primal variable preserves local conservation we prove existence and uniqueness of a solution for the mhm method as well as optimal convergence estimates of any order in the natural norms also we propose a face residual a posteriori error estimator and prove that it controls the error of both variables in the natural norms several numerical tests assess the theoretical results

discontinuous petrov galerkin method with optimal test functions for thin body problems in solid mechanics we study the applicability of the discontinuous petrov galerkin dpg variational framework for thin body problems in structural mechanics our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate local computation of the corresponding optimal test functions in the timoshenko beam problem the proposed method is shown to provide the best approximation in an energy type norm which is equivalent to the l2 norm for all the unknowns uniformly with respect to the thickness parameter the same formulation remains valid also for the asymptotic euler bernoulli solution as another one dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations in particular we derive a special norm for the test space which leads to a robust method in terms of the shell thickness finally we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp mesh for resolving the boundary layer

dpg method with optimal test functions for a transmission problem we propose and analyze a numerical method to solve an elliptic transmission problem in full space the method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra weak formulation in the interior to guarantee the discrete inf sup condition the system is discretized by the dpg method with optimal test functions we prove that the principal unknowns are approximated quasi optimally numerical experiments for problems with smooth and singular solutions confirm optimal convergence orders

a posteriori error control for dpg methods a combination of ideas in least squares finite element methods with those of hybridized methods recently led to discontinuous petrov galerkin dpg finite element methods they minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard locally computable dual norm this paper establishes a general a posteriori error analysis for the natural norms of the dpg schemes under conditions equivalent to a priori stability estimates it is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors the presented abstract framework for a posteriori error analysis applies to known dpg discretizations of laplace and lame equations and to a novel dpg method for the stress velocity formulation of stokes flow with symmetric stress approximations since the error control does not rely on the discrete equations it applies to inexactly computed or otherwise perturbed solutions within the discrete space

computational aspects of the ultra weak variational formulation the ultra weak variational formulation uwvf approach has been proposed as an effective method for solving helmholtz problems with high wave numbers however for coarse meshes the method can suffer from instability in this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous helmholtz problem we introduce a method to improve the uwvf scheme and we compare iterative solvers for the resulting linear system computations for the acoustic transmission problem in 2d show that the new approach enables helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers

an unconditionally stable space time fe method for the korteweg de vries equation abstract we introduce an unconditionally stable finite element fe method the automatic variationally stable fe avs fe method for the numerical analysis of the korteweg de vries kdv equation the avs fe method is a petrov galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous petrov galerkin dpg method however since avs fe method is a minimum residual method we establish a global saddle point system instead of computing optimal test functions element by element this system allows us to seek both the approximate solution of the kdv initial boundary value problem ibvp and a riesz representer of the approximation error the avs fe method distinguishes itself from other minimum residual methods by using globally continuous hilbert spaces such as h 1 while at the same time using broken hilbert spaces for the test consequently the avs fe approximations are classical c 0 continuous fe solutions the unconditional stability of this method allows us to solve the kdv equation space and time without having to satisfy a cfl condition we present several numerical verifications for both linear and nonlinear versions of the kdv equation leading to optimal convergence behavior finally we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear kdv equation

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

a collocated isogeometric finite element method based on gauss lobatto lagrange extraction of splines generalizing the concept of bezier extraction we introduce an extraction operator that links c0c0 gauss lobatto lagrange functions with smooth splines this opens the door for collocated isogeometric analysis that combines the accuracy of the galerkin method with collocation type formation and assembly procedures we present the key ingredients of the technology i e integration by parts and the weighted residual form the interaction of gauss lobatto lagrange extraction with gauss lobatto quadrature and symmetrization with the ultra weak formulation we compare the new method with standard isogeometric galerkin and isogeometric point collocation methods for spline discretizations in three dimensions

calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media we present an analytical and numerical investigation of the finite difference computation of the equivalent conductivity of heterogeneous porous media the customary harmonic scheme to evaluate finite difference internodal transmissivities produces a systematic bias in the numerical results unless an extremely fine grid is used in order to quantify such effects we have developed an analytical approach in the form of a series expansion of the equivalent numerical conductivity in powers of the conductivity variance this leads to an expression of the numerical answer as a function of the grid block size the calculation confirms the existence of a strong bias and of a very slow convergence we propose a simple method to correct it which is well suited for upscaling numerical experiments performed with more contrasting heterogeneous media show similar results this allows the use of coarser gridding and consequently an appreciable speedup in the numerical simulation approach

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

note on discontinuous trace approximation in the practical dpg method we analyze a non conforming dpg method with discontinuous trace approximation for the poisson problem in two and three space dimensions we show its well posedness and quasi optimal convergence in the principal unknown numerical experiments confirming the theory have been presented previously

a family of multiscale hybrid mixed finite element methods for the darcy equation with rough coefficients abstract we aim at proposing novel stable finite element methods for the mixed darcy equation with heterogeneous coefficients within a space splitting framework we start from the primal hybrid formulation of the elliptic model for the pressure localization of this infinite dimensional problem leads to element level boundary value problems which embed multiscale and high contrast features in a natural way with neumann boundary conditions driven by the lagrange multipliers such a procedure leads to methods involving the space of piecewise constants for the pressure together with a discretization of the fluxes choosing arbitrarily polynomial interpolations the lowest order raviart thomas element as well as some recent multiscale methods are recovered in addition the methods assure local mass conservation and can be interpreted as stabilized primal hybrid methods extensive numerical validation attests to the accuracy of the new methods on academic and more realistic problems with rough coefficients

a primal dpg method without a first order reformulation abstract we show that it is possible to apply the dpg methodology without reformulating a second order boundary value problem into a first order system by considering the simple example of the poisson equation the result is a new weak formulation and a new dpg method for the poisson equation which has no numerical trace variable but has a numerical flux approximation on the element interfaces in addition to the primal interior variable

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

analysis of non conforming dpg methods on polyhedral meshes using fractional sobolev norms abstract the work is concerned with two problems a analysis of a discontinuous petrov galerkin dpg method set up in fractional energy spaces b use of the results to investigate a non conforming version of the dpg method for general polyhedral meshes we use the ultraweak variational formulation for the model laplace equation the theoretical estimates are supported with 3d numerical experiments

a symmetric nodal conservative finite element method for the darcy equation this work introduces and analyzes novel stable petrov galerkin enriched methods pgem for the darcy problem based on the simplest but unstable continuous mathbb p 1 mathbb p 0 pair stability is recovered inside a petrov galerkin framework where elementwise dependent residual functions named multiscale functions enrich both velocity and pressure trial spaces unlike the velocity test space that is augmented with bubble like functions multiscale functions correct edge residuals as well the multiscale functions turn out to be the well known lowest order raviart thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure the enrichment strategy suggests the way to recover the local mass conservation property for nodal based interpolation spaces we prove that the method and its symmetric version are well posed and achieve optimal error estimates in natural norms numerical validations confirm claimed theoretical results