coupling of discontinuous galerkin schemes for viscous flow in porous media with adsorption polymer flooding is an important stage of enhanced oil recovery in petroleum reservoir engineering a model of this process is based on the study of multicomponent viscous flow in porous media with adsorption this model can be expressed as a brinkman based model of flow in porous media coupled to a nonstrictly hyperbolic system of conservation laws for multiple components water and polymers that form the aqueous phase the discretization proposed for this coupled flow transport problem combines an h div conforming discontinuous galerkin dg method for the brinkman flow problem with a classical dg method for the transport equations the dg formulation of the transport problem is based on discontinuous numerical fluxes an invariant region property is proved under the mild assumption that the underlying mesh is a b triangulation b cockburn s hou and c w shu math comp 54 1990 pp 545 581 this property states that only physically relevant bounded and nonnegative saturation and

the invertibility of rotation invariant radon transforms let r denote the radon transform on rn that integrates a function over hyperplanes in given smooth positive measures depending on the hyperplane we characterize the measures for which r is rotation invariant we prove rotation invariant transforms are all one to one and hence invertible on the domain of square integrable functions of compact support l02 rn we prove the hole theorem if f l02rn and r f 0 for hyperplanes not intersecting a ball centered at the origin then f is zero outside of that ball using the theory of fourier integral operators we extend these results to the domain of distributions of compact support on rn our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated radon transform as well as invertibility for other radon transforms

on the asymptotic exactness of error estimators for linear triangular finite elements this paper deals with a posteriori error estimates for piecewise linear finite element approximations of elliptic problems we analyze two estimators based on recovery operators for the gradient of the approximate solution by using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution one of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation this estimator is equivalent to the error in the energy norm under rather general conditions however we show that for the asymptotic exactness the regularity assumption on the mesh is not merely technical while doing this we analyze the relation between superconvergence and asymptotic exactness for some particular examples

stationary sets for the wave equation in crystallographic domains let w be a crystallographic group in r n generated by reflections and let be the fundamental domain of w we characterize stationary sets for the wave equation in when the initial data is supported in the interior of the stationary sets are the sets of time invariant zeros of nontrivial solutions that are identically zero at t 0 lie show that for these initial data the n 1 dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group w w w this part comes from a corresponding odd symmetry of the initial data in physical language the result is that if the initial source is localized strictly inside of the crystalline then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one

a new paradigm for parallel adaptive meshing algorithms we present a new approach to the use of parallel computers with adaptive finite element methods this approach addresses the load balancing problem in a new way requiring far less communication than current approaches it also allows existing sequential adaptive pde codes such as pltmg and mc to run in a parallel environment without a large investment in recoding in this new approach the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor using a sequential adaptive solver without requiring any modifications to the sequential solver the small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh which are then used in a weighted recursive spectral bisection of the initial mesh the bulk of the calculation then takes place independently on each processor with no communication using possibly the same sequential adaptive solver each processor adapts its region of the mesh independently and a nearly load balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection only the initial fan out of the mesh decomposition to the processors requires communication two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution namely the construction of a global conforming mesh from the independent subproblems followed by a final smoothing phase using the subdomain solutions as an initial guess we present a series of convincing numerical experiments which illustrate the effectiveness of this approach the justification of the initial refinement prediction step as well as the justification of skipping the two communication intensive steps is supported by some recent j xu and a zhou math comp to appear and not so recent j a nitsche and a h schatz math comp 28 1974 pp 937 958 a h schatz and l b wahlbin math comp 31 1977 pp 414 442 a h schatz and l b wahlbin math comp 64 1995 pp 907 928 results on local a priori and a posteriori error estimation

mixed displacement rotation pressure formulations for linear elasticity abstract we propose a new locking free family of mixed finite element and finite volume element methods for the approximation of linear elastostatics formulated in terms of displacement rotation vector and pressure the unique solvability of the three field continuous formulation as well as the well definiteness and stability of the proposed galerkin and petrov galerkin methods is established thanks to the babuska brezzi theory optimal a priori error estimates are derived using norms robust with respect to the lame constants turning these numerical methods particularly appealing for nearly incompressible materials we exemplify the accuracy in a suitably weighted norm as well the applicability of the new formulation and the mixed schemes by conducting a number of computational tests in 2d and 3d also including cases not covered by our theoretical analysis

a posteriori error estimation for an augmented mixed primal method applied to sedimentation consolidation systems abstract in this paper we develop the a posteriori error analysis of an augmented mixed primal finite element method for the 2d and 3d versions of a stationary flow and transport coupled system typically encountered in sedimentation consolidation processes the governing equations consist in the brinkman problem with concentration dependent viscosity written in terms of cauchy pseudo stresses and bulk velocity of the mixture coupled with a nonlinear advection nonlinear diffusion equation describing the transport of the solids volume fraction we derive two efficient and reliable residual based a posteriori error estimators for a finite element scheme using raviart thomas spaces of order k for the stress approximation and continuous piecewise polynomials of degree k 1 for both velocity and concentration for the first estimator we make use of suitable ellipticity and inf sup conditions together with a helmholtz decomposition and the local approximation properties of the clement interpolant and raviart thomas operator to show its reliability whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle bubble and edge bubble functions next we analyse an alternative error estimator whose reliability can be proved without resorting to helmholtz decompositions finally we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed primal finite element method

a residual based a posteriori error estimator for the stokes darcy coupled problem in this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow the flows are governed by the stokes and darcy equations respectively and the transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the finite element subspaces consider bernardi raugel and raviart thomas elements for the velocities piecewise constants for the pressures and continuous piecewise linear elements for a lagrange multiplier defined on the interface we derive a reliable and efficient residual based a posteriori error estimator for this coupled problem the proof of reliability makes use of suitable auxiliary problems diverse continuous inf sup conditions satisfied by the bilinear forms involved and local approximation properties of the clement interpolant and raviart thomas operator on the other hand helmholtz decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are the main tools for proving the efficiency of the estimator up to minor modifications our analysis can be extended to other finite element subspaces yielding a stable galerkin scheme

mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media abstract in this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media the proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise then discretise approach for the approximation of the optimal control problem leading to nonsymmetric algebraic systems and employing minimum regularity requirements estimates for the error between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed discontinuous approximation measured in suitable norms are derived showing optimal orders of convergence

a technique for realizing linear phase iir filters a real time iir filter structure is presented that possesses exact phase linearity with 10 approximately 1000 times fewer general multiplies than conventional fir filters of similar performance and better magnitude characteristics than equiripple or maximally flat group delay iir filters this structure is based on a technique using local time reversal and single pass sectioned convolution methods to realized a real time recursive implementation of the noncausal transfer function h z sup 1 the time reversed section technique used to realize exactly linear phase iir filters is described the effects of finite section length on the sectional convolution are analyzed a simulation methodology is developed to address the special requirements of simulating a time reversed section filter a design example is presented with computer simulation to illustrate performance in terms of overall magnitude response and phase linearity as a function of finite section length nine example filter specifications are used to compare the performance and complexity of the time reversed section technique to those of a direct fir implementation

injectivity sets for the radon transform over circles and complete systems of radial functions abstract a necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane this problem is shown to be equivalent to inversion for the radon transform on circles centered on restricted subsets of the plane the proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular radon transform a characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems and questions of pinkus and ehrenpreis are answered

a synchronous and iterative flux correction formalism for coupled transport equations many problems of fluid dynamics involve the coupled transport of several density like dependent variables for instance densities of mass and momenta in elastic flows in this paper a conservative and synchronous flux corrected transport fct formalism is developed which aims at a consistent transport of such variables the technique differs from traditional fct algorithms in two respects first the limiting of transportive fluxes of the primary variables e g mass and momentum does not derive from smooth estimates of the variables but it derives from analytic constraints implied by the lagrangian form of the governing continuity equations which are imposed on the specific mixing ratios of the variables e g velocity components second the traditional fct limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements this procedure can also be used in the framework of traditional fct schemes and a demonstration is provided that it can significantly reduce some of the pathological behaviors of fct algorithms although the approach derived is applicable to the transport of arbitrary conserved quantities it is particularly useful for the synchronous transport of mass and momenta in elastic flows where it assures intrinsic stability of the algorithm regardless of the magnitude of the mass density variable this latter property becomes especially important in fluids with large density variations or in models with a material vertical coordinate e g geophysical hydrostatic stratified flows in isopycnic isentropic coordinates where material surfaces can collapse to zero mass layers admitting therefore arbitrarily large local courant numbers

a crank nicolson discontinuous finite volume element method for a coupled non stationary stokes darcy problem abstract in this paper we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a crank nicolson discretization in time and we obtain the solution at the first time step by applying a first order backward euler method to solve the coupling of time dependent stokes darcy problem the proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head whereas the pressure is approximated by the piecewise constant elements then the unique solvability of the approximate solution for the discrete problem is derived in addition the optimal error estimates of the full discretization in standard l 2 norm and broken h 1 norm are obtained finally a series of numerical experiments are provided to illustrate the features of the proposed method such as the optimal accuracy orders adaptive mesh refinement mass conservation capability to conveniently deal with complicated geometries with different types of boundary conditions and the applicability to the problems with realistic parameters in particular the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain it encompasses both fractures with high permeability which act as the prior flow conduit and fractures with low permeability as a flow barrier

analysis of a pseudostress based mixed finite element method for the brinkman model of porous media flow in this paper we introduce and analyze a new mixed finite element method for the two dimensional brinkman model of porous media flow with mixed boundary conditions we use a dual mixed formulation in which the main unknown is given by the pseudostress the original velocity and pressure unknowns are easily recovered through a simple postprocessing in addition since the neumann boundary condition becomes essential we impose it in a weak sense which yields the introduction of the trace of the fluid velocity over the neumann boundary as the associated lagrange multiplier we apply the babuska brezzi theory to establish sufficient conditions for the well posedness of the resulting continuous and discrete formulations in particular a feasible choice of finite element subspaces is given by raviart thomas elements of order k ge 0 k 0 for the pseudostress and continuous piecewise polynomials of degree k 1 k 1 for the lagrange multiplier we also derive a reliable and efficient residual based a posteriori error estimator for this problem suitable auxiliary problems the continuous inf sup conditions satisfied by the bilinear forms involved a discrete helmholtz decomposition and the local approximation properties of the raviart thomas and clement interpolation operators are the main tools for proving the reliability then helmholtz s decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are employed to show the efficiency finally several numerical results illustrating the performance and the robustness of the method confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithm are provided

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

a posteriori error estimates for the stokes problem an a posteriors error estimate is derived and analyzed for the mini element discretization of the stokes equations the estimate is based on the solution of a local stokes problem in each element of the finite element mesh using spaces of quadratic bump functions for both velocity and pressure errors this results in solving a 9 times 9 system which reduces to two easily invertible 3 times 3 systems comparisons with other estimates based on a petrov galerkin solution are used in this analysis which shows that it provides a reasonable approximation of the actual discretization error numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures

generalized difference methods for a nonlinear dirichlet problem the paper is devoted to the construction of generalized difference methods on irregular meshes for second order nonlinear elliptic equations based on the generalized galerkin method the error estimate is proved

phase improvement in iir digital filters a new technique is presented for improving the phase response of finite impulse response iir digital filters in this technique a measure of phase nonlinearity is developed given an iir filter which has the desired amplitude response and it realized as a cascade of second order sections the measure is minimized by making some of the sections anticausal efficient implementation of this algorithm is discussed examples illustrating the technique are also given

analysis of a finite volume element method for the stokes problem in this paper we propose a stabilized conforming finite volume element method for the stokes equations on stating the convergence of the method optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations a superconvergence result is also derived by using a postprocessing projection method in particular the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries finally some numerical experiments that confirm the predicted behavior of the method are provided

maximum norm a posteriori error estimates for a one dimensional convection diffusion problem a singularly perturbed quasi linear two point boundary value problem with an exponential boundary layer is discretized on arbitrary nonuniform meshes using first and second order difference schemes including upwind schemes we give first and second order maximum norm a posteriori error estimates that are based on difference derivatives of the numerical solution and hold true uniformly in the small parameter numerical experiments support the theoretical results

a posteriori error estimates for mixed fem in elasticity a residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear planar elasticity the proof of the reliability of the estimator is based on helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements the efficiency follows from inverse estimates the constants in both estimates are independent of the lame constant lambda and so locking phenomena for lambda to infty are properly indicated the analysis justifies a new adaptive algorithm for automatic mesh refinement

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

efficient time stepping methods for miscible displacement problems in porous media efficient procedures for time stepping galerkin methods for approximating the solution of a coupled system for c c x t and p p x t with nonlinear neumann boundary conditions of the form begin gathered nabla cdot a x c nabla p y x c nabla g equiv nabla cdot u f 1 x t quad x in omega quad t in 0 t hfill nabla cdot b x c nabla p nabla c u cdot nabla c phi x frac partial c partial t f 2 x t c quad x in omega quad t in 0 t hfill u cdot v q 1 x t quad x in partial omega quad t in 0 t hfill b frac partial c partial v q x t c quad x in partial omega quad t in 0 t hfill c x 0 c 0 x quad x in omega hfill end gathered where omega subset mathbb r d 2 leqq d leqq 3 are presented and analyzed this system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions a

an a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem a simple algorithm for adaptive timestep control is presented for a backward euler discretization of a linear parabolic problem the algorithm is based on an a posteriori error estimate involving the computed approximate solution it is proved that with only very rough a priori information on the exact solution the algorithm will choose a sequence of timesteps for which the error will be controlled up to a constant uniformly in time on a given tolerance level

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

discontinuous approximation of viscous two phase flow in heterogeneous porous media runge kutta discontinuous galerkin rkdg and discontinuous finite volume element dfve methods are applied to a coupled flow transport problem describing the immiscible displacement of a viscous incompressible fluid in a non homogeneous porous medium the model problem consists of nonlinear pressure velocity equations assuming brinkman flow coupled to a nonlinear hyperbolic equation governing the mass balance saturation equation the mass conservation properties inherent to finite volume based methods motivate a dfve scheme for the approximation of the brinkman flow in combination with a rkdg method for the spatio temporal discretization of the saturation equation the stability of the uncoupled schemes for the flow and for the saturation equations is analyzed and several numerical experiments illustrate the robustness of the numerical method

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

a posteriori error estimators for the stokes equations ii non conforming discretizations we present an a posteriori error estimator for the non conforming crouzeix raviart discretization of the stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation the error estimator yields global upper and local lower bounds for the error of the finite element solution it can easily be generalized to the stationary incompressible navier stokes equations and to other non conforming finite element methods numerical examples show the efficiency of the proposed error estimator

analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements this paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method these bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes they depend strongly on the geometry of the triangles and relatively weakly on the smoothness of the solution an example shows that the bounds are not over pessimistic in babuska plank and rodriguez 4 detailed numerical experimentation is given

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

analysis of a velocity pressure pseudostress formulation for the stationary stokes equations abstract we consider a non standard mixed approach for the stokes problem in which the velocity the pressure and the pseudostress are the main unknowns alternatively the pressure can be eliminated from the original equations thus yielding an equivalent formulation with only two unknowns in this paper we develop a priori and a posteriori error analyses of both approaches we first apply the babuska brezzi theory to prove the well posedness of the continuous and discrete formulations in particular we show that raviart thomas elements of order k 0 for the pseudostresses and piecewise polynomials of degree k for the velocities and the pressures ensure unique solvability and stability of the associated galerkin schemes then we derive reliable and efficient residual based a posteriori error estimators for both schemes without and with pressure unknown finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimators and showing the behaviour of the associated adaptive algorithms

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

the khavinson shapiro conjecture and polynomial decompositions the main result of the paper states the following let be a polynomial in n variables of degree t suppose that there exists a constant c 0 such that any polynomial f has a polynomial decomposition f qf hf with k hf 0 and degqf degf c then deg 2k here k is the kth iterate of the laplace operator as an application new classes of domains in r n are identied for which the khavinson shapiro conjecture holds

error estimates for discontinuous galerkin method for nonlinear parabolic equations abstract we consider the nonlinear parabolic partial differential equations we construct a discontinuous galerkin approximation using a penalty term and obtain an optimal l l 2 error estimate

galerkin least squares finite element method for the obstacle problem we construct a consistent multiplier free method for the finite element solution of the obstacle problem the method is based on an augmented lagrangian formulation in which we eliminate the multip

a discontinuous mixed covolume method for elliptic problems we develop a discontinuous mixed covolume method for elliptic problems on triangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

morera theorems for complex manifolds abstract we prove morera theorems for the radon transform integrating on geodesic spheres on complex analytic manifolds of arbitrary dimension to avoid pathologies we assume that the radius of each sphere of integration is less than the injectivity radius at its center the proofs of the main results are local and they involve the microlocal properties of associated radon transforms and a theorem of hormander kawai and kashiwara on microlocal singularities we consider morera theorems for spheres of fixed radius and spheres of arbitrary radius

on nodal sets for dirac and laplace operators we prove that the nodal set zero set of a solution of a generalized dirac equation on a riemannian manifold has codimension 2 at least if the underlying manifold is a surface then the nodal set is discrete we obtain a quick proof of the fact that the nodal set of an eigenfunction for the laplace beltrami operator on a riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension we also see that the nodal set of a harmonic differential form on a closed manifold has codimension 2 at least a fact which is not true if the manifold is not closed examples show that all bounds are optimal

a priori and a posteriori error analyses of a velocity pseudostress formulation for a class of quasi newtonian stokes flows abstract in this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear stokes models arising in quasi newtonian fluids the methods are based on a non standard mixed approach in which the velocity the pressure and the pseudostress are the original unknowns however we use the incompressibility condition to eliminate the pressure and set the velocity gradient as an auxiliary unknown which yields a twofold saddle point operator equation as the resulting dual mixed variational formulation in addition a suitable augmented version of the latter showing a single saddle point structure is also considered we apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well posed in particular we show that raviart thomas elements of order k 0 for the pseudostress and piecewise polynomials of degree k for the velocity and its gradient ensure the well posedness of the associated galerkin schemes moreover we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation then we derive a reliable and efficient residual based a posteriori error estimator for each scheme finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithms

hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems in this paper we have analyzed a one parameter family of hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems nabla cdot rm a u nabla u f u nabla u 0 with dirichlet boundary conditions these methods depend on the values of the parameter theta in 1 1 where 1 corresponds to the nonsymmetric and 1 corresponds to the symmetric interior penalty methods when rm a u nabla u nabla u and f u u f that is for the poisson problem the error estimate in the broken h 1 norm which is optimal in h mesh size and suboptimal in p degree of approximation is derived using piecewise polynomials of degree p 2 when the solution u in h 5 2 omega in the case of linear elliptic problems also this estimate is optimal in h and suboptimal in p further optimal error estimate in the l 2 norm when 1 is derived numerical experiments are presented to illustrate the theoretical results

an adaptive finite element strategy for the three dimensional time dependent navier stokes equations abstract an adaptive strategy for three dimensional time dependent problems in the context of the fem is presented the basic tools are a mechanism for local refinement and coarsening of simplicial meshes and an unexpensive error estimator the algorithm for local grid modification is based on bisecting tetrahedra the method is applied to the navier stokes equations

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 2 a posteriori error estimation abstract this is the second paper out of a series of papers devoted to model and hpq adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description in our previous publication we investigated the issue of hierarchical models and approximations of such structures we applied 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within the proposed finite element formulation furthermore we assumed that the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q could vary locally i e they could be different in each finite element the a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints the generalized method is applied to the assessment of the total and approximation errors while the modeling error is calculated as the difference between the former two errors the corresponding error controlled adaptive procedures are based on a three step strategy with possible iterations of h and p steps

linear phase iir filter with equiripple stopband a simple technique of introducing transmission zeros into a low pass all pole filter function to make its stopband transmission equiripple is described the zeros are introduced through a transformation which maps the stopband of the filter onto the unit circle and the passband onto a segment of the real axis in the transformed plane the technique is used to improve the loss characteristic of all pole constant delay filters examples are included to illustrate the design procedure

discontinuous finite volume element discretization for coupled flow transport problems arising in models of sedimentation the sedimentation consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the stokes problem written in terms of the mixture flow velocity and the pressure field here both the viscosity and the forcing term depend on the local solids concentration a semi discrete discontinuous finite volume element dfve scheme is proposed for this model the numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field whereas the pressure is approximated by piecewise constant elements the unique solvability of both the nonlinear continuous problem and the semi discrete dfve scheme is discussed and optimal convergence estimates in several spatial norms are derived properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples including flows under gravity with changing direction a secondary settling tank in an axisymmetric setting and batch sedimentation in a tilted cylindrical vessel

a stabilized finite volume element formulation for sedimentation consolidation processes a model of sedimentation consolidation processes in so called clarifier thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the stokes system for an incompressible fluid describing the motion of the mixture in cylindrical coordinates and if an axially symmetric solution is assumed the original problem reduces to two space dimensions this poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted sobolev spaces a novel finite volume element method is introduced for the spatial discretization where the velocity field and the solids concentration are discretized on two different dual meshes the method is based on a stabilized discontinuous galerkin formulation for the concentration field and a multiscale stabilized pair of mathbb p 1 mathbb p 1 elements for velocity and pressure respectively numerical experiments illustrate properties of the model and the satisfactory p

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

unified analysis of finite volume methods for the stokes equations we establish a general framework for analyzing the class of finite volume methods for the stokes equations under the framework optimal order convergence for velocity and pressure in various sobolev norms are obtained in a natural and systematic way specifically optimal l 2 error estimates for velocity are obtained for the first time for several different finite volume methods

a flux corrected transport algorithm for handling the close packing limit in dense suspensions convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level in this paper we are concerned with solving continuity equations that govern the evolution of volume fractions in eulerian models of disperse two phase flows an implicit galerkin finite element approximation is equipped with a flux limiter for the convective terms the fully multidimensional limiting strategy is based on a flux corrected transport fct algorithm this nonlinear high resolution scheme satisfies a discrete maximum principle for divergence free velocities and ensures positivity preservation for arbitrary velocity fields to enforce an upper bound that corresponds to the maximum packing limit an fct like overshoot limiter is applied to the converged convective fluxes at the end of each time step this postprocessing step imposes an additional physical constraint on the numerical solution to the unconstrained mathematical model numerical results for 2d implosion problems illustrate the performance of the proposed limiting procedure

a conservative and monotone mixed hybridized finite element approximation of transport problems in heterogeneous domains in this article we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties the model in each subdomain consists of a partial differential equation with diffusive convective and reactive terms the coupling between each subdomain being realized through an interface transmission condition of robin type the numerical approximation of the problem in the two dimensional case is carried out through a dual mixed hybridized finite element method with numerical quadrature of the mass flux matrix the resulting method is a conservative finite volume scheme over triangular grids for which a discrete maximum principle is proved under the assumption that the mesh is of delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface the stability accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology electrophysiology and neuroelectronics

a banach spaces based analysis of a new mixed primal finite element method for a coupled flow transport problem abstract in this paper we introduce and analyze a new finite element method for a strongly coupled flow and transport problem in r n n 2 3 whose governing equations are given by a scalar nonlinear convection diffusion equation coupled with the stokes equations the variational formulation for this model is obtained by applying a suitable dual mixed method for the stokes system and the usual primal procedure for the transport equation in this way and differently from the techniques previously developed for this and related coupled problems no augmentation procedure needs to be incorporated now into the solvability analysis which constitutes the main advantage of the present approach the resulting continuous and discrete schemes which involve the cauchy fluid stress the velocity of the fluid and the concentration as the only unknowns are then equivalently reformulated as fixed point operator equations consequently the well known schauder banach and brouwer theorems combined with babuska brezzi s theory in banach spaces monotone operator theory regularity assumptions and sobolev imbedding theorems allow to establish the corresponding well posedness of them in particular raviart thomas approximations of order k 0 for the stress discontinuous piecewise polynomials of degree k for the velocity and continuous piecewise polynomials of degree k 1 for the concentration becomes a feasible choice for the galerkin scheme next suitable strang type lemmas are employed to derive optimal a priori error estimates finally several numerical results illustrating the performance of the mixed primal scheme and confirming the theoretical rates of convergence are provided

iir digital filters with equiripple stopband transmission recently a simple technique of introducing transmission zeros into a low pass all pole filter transfer function to make its stop band transmission equiripple has been described by hazra 1 it is shown that this technique represents a special case of a more general procedure which provides not only hazra s solution but also additional filters which are advantageous with regard to the required number of multiplications per sample the results are demonstrated with the aid of an example

convergence of simple adaptive galerkin schemes based on h h 2 error estimators we discuss several adaptive mesh refinement strategies based on h h 2 error estimation this class of adaptive methods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead we prove that under the saturation assumption these adaptive algorithms are convergent our framework applies not only to finite element methods but also yields a first convergence proof for adaptive boundary element schemes for a finite element model problem we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general

a discontinuous finite volume method for the stokes problems we develop a new discontinuous finite volume method for solving the stokes equations on both triangular and rectangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

partitioned coupling of advection diffusion reaction systems and brinkman flows we present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection diffusion reaction systems and incompressible viscous flow the space discretisation of the governing equations is based on mixed finite element methods defined on unstructured meshes whereas the time integration hinges on an operator splitting strategy that exploits the differences in scales between the reaction advection and diffusion processes considering the global system as a number of sequentially linked sets of partial differential and algebraic equations the flow solver presents the advantage that all unknowns in the system here vorticity velocity and pressure can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective the robustness of the proposed method is illustrated with a series of numerical tests in 2d and 3d relevant in the modelling of bacterial bioconvection and boussinesq systems

algorithms for refining triangular grids suitable for adaptive and multigrid techniques two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed the algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used the algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way proofs of the above properties are presented the second algorithm is a simpler improved version of the first which retains most of the properties of the latter the algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation in this sense they can be adequately combined with adaptive and or multigrid techniques for solving finite element systems examples of the application of the algorithms are given and two possible generalizations are pointed out

mixed finite element discontinuous finite volume element discretization of a general class of multicontinuum models we present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua in the framework of mixture theory we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases the model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture barycentric velocities phase pressure and saturation we construct an accurate robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume element discretization of the mass or transport equations the scheme is devised for unstructured meshes and relies on mixed brezzi douglas marini approximations of phase and total velocities on piecewise constant elements for the approximation of phase or total pressures as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest such as volume fraction total density saturation etc as the discretization scheme is derived for a general formulation of multicontinuum physical systems it can be readily applied to a large class of simplified multiphase models on the other the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate an extensive set of numerical test cases involving two and three dimensional porous media are presented to demonstrate the accuracy of the method displaying an optimal convergence rate the physics preserving properties of the mixed primal scheme as well as the robustness of the method which is successfully used to simulate diverse physical phenomena such as density fingering terzaghi s consolidation deformation of a cantilever bracket and boycott effects the applicability of the method is not limited to flow in porous media but can also be employed to describe many other physical systems governed by a similar set of equations including e g multi component materials

a new discontinuous finite volume method for elliptic problems we develop and analyze a new discontinuous finite volume method for second order elliptic problems with tensor coefficients an optimal order error estimate is obtained in a mesh dependent norm an l2 error estimate is also obtained

a priori and a posteriori error analyses of a pseudostress based mixed formulation for linear elasticity in this paper we present the a priori and a posteriori error analyses of a non standard mixed finite element method for the linear elasticity problem with non homogeneous dirichlet boundary conditions more precisely the approach introduced here is based on a simplified interpretation of the pseudostress displacement formulation originally proposed in arnold and falk 1988 which does not require symmetric tensor spaces in the finite element discretization in addition physical quantities such as the stress the strain tensor of small deformations and the rotation are computed through a simple postprocessing in terms of the pseudostress variable furthermore we also introduce a second element by element postprocessing formula for the stress which yields an optimally convergent approximation of this unknown with respect to the broken h div norm we apply the classical babuska brezzi theory to prove that the corresponding continuous and discrete schemes are well posed in particular raviart thomas spaces of order k 0 for the pseudostress and piecewise polynomials of degree k for the displacement can be utilized moreover we remark that in the 3d case the number of unknowns behaves approximately as 9 times the number of elements tetrahedra of the triangulation when k 0 this factor increases to 12 5 when one uses the classical peers next we derive a reliable and efficient residual based a posteriori error estimator for the mixed finite element scheme finally several numerical results illustrating the performance of the method confirming the theoretical properties of the estimator and showing the expected behaviour of the associated adaptive algorithm are provided

range descriptions for the spherical mean radon transform abstract the transform considered in the paper averages a function supported in a ball in r n over all spheres centered at the boundary of the ball this radon type transform arises in several contemporary applications e g in thermoacoustic tomography and sonar and radar imaging range descriptions for such transforms are important in all these areas for instance when dealing with incomplete data error correction and other issues four different types of complete range descriptions are provided some of which also suggest inversion procedures necessity of three of these appropriately formulated conditions holds also in general domains while the complete discussion of the case of general domains would require another publication

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m