robust a posteriori error estimation for finite element approximation to h curl problem abstract in this paper we introduce a novel a posteriori error estimator for the conforming finite element approximation to the h curl problem with inhomogeneous media and with the right hand side only in l 2 the estimator is of the recovery type independent with the current approximation to the primary variable the electric field an auxiliary variable the magnetizing field is recovered in parallel by solving a similar h curl problem an alternate way of recovery is presented as well by localizing of the error flux the estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable it is proved that the estimator is approximately equal to the true error in the energy norm without the quasi monotonicity assumption finally we present numerical results for several h curl interface problems

a note on constant free a posteriori error estimates in this note we look at constant free a posteriori error estimates from a different perspective we show that they can be interpreted as an alternative way of expressing the residual of a finite element approximation and thus fit into the same framework as other a posteriori error estimates such as residual error indicators our approach also reveals that when applied to singularly perturbed reaction diffusion or convection diffusion problems constant free a posteriori error estimates will not be fully robust unless extra measures are taken

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

polynomial degree robust a posteriori estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed discretizations we present equilibrated flux a posteriori error estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed finite element discretizations of the two dimensional poisson problem relying on the equilibration by the mixed finite element solution of patchwise neumann problems the estimates are guaranteed locally computable locally efficient and robust with respect to polynomial degree maximal local overestimation is guaranteed as well numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous galerkin scheme

a priori and posteriori error analysis for time dependent maxwell s equations abstract we consider time dependent maxwell s equations discretized by variable time steps in time domain and edge elements in spatial domain first the stability and optimal a priori error estimate are proved for both semi and fully discrete schemes then a posteriori error analysis is carried out for both schemes

goal oriented error estimation based on equilibrated flux reconstruction for finite element approximations of elliptic problems abstract we propose an approach for goal oriented error estimation in finite element approximations of second order elliptic problems that combines the dual weighted residual method and equilibrated flux reconstruction methods for the primal and dual problems the objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution it is only assumed here that the discretization methods come with a priori error estimates and an equilibrated flux reconstruction algorithm a high order discontinuous galerkin dg method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes one contribution of the paper is to show how the order of the dg method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method a dg method or a mixed raviart thomas method is used for the solution of the primal problem numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size

two guaranteed equilibrated error estimators for harmonic formulations in eddy current problems abstract in this paper a guaranteed equilibrated error estimator is developed for the 3d harmonic magnetodynamic problem of maxwell s system this system is recasted in the classical a potential formulation and solved by the finite element method the error estimator is built starting from the a numerical solution by a local flux reconstruction technique its equivalence with the error in the energy norm is established a comparison of this estimator with an equilibrated error estimator already developed through a complementary problem points out the advantages and drawbacks of these two estimators in particular an analytical benchmark test illustrates the obtained theoretical results and a physical benchmark test shows the efficiency of these two estimators

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

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a robust and accurate approach to computing compressible multiphase flow stratified flow model and ausm up scheme in this paper we propose a new approach to compute compressible multifluid equations firstly a single pressure compressible multifluid model based on the stratified flow model is proposed the stratified flow model which defines different fluids in separated regions is shown to be amenable to the finite volume method we can apply the conservation law to each subregion and obtain a set of balance equations secondly the ausm scheme which is originally designed for the compressible gas flow is extended to solve compressible liquid flows by introducing additional dissipation terms into the numerical flux the new scheme called ausm up can be applied to both liquid and gas flows thirdly the contribution to the numerical flux due to interactions between different phases is taken into account and solved by the exact riemann solver we will show that the proposed approach yields an accurate and robust method for computing compressible multiphase flows involving discontinuities such as shock waves and fluid interfaces several one dimensional test problems are used to demonstrate the capability of our method including the ransom s water faucet problem and the air water shock tube problem finally several two dimensional problems will show the capability to capture enormous details and complicated wave patterns in flows having large disparities in the fluid density and velocities such as interactions between water shock wave and air bubble between air shock wave and water column s and underwater explosion

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

robust error bounds for finite element approximation of reaction diffusion problems with non constant reaction coefficient in arbitrary space dimension abstract we present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction diffusion problems with mixed boundary conditions and piecewise constant reaction coefficient formulated in arbitrary dimension the estimator provides a guaranteed upper bound on the energy norm of the error and it is robust for all values of the reaction coefficient including the singularly perturbed case the approach is based on robustly equilibrated boundary flux functions of ainsworth and oden 2000 and on subsequent robust and explicit flux reconstruction this paper simplifies and extends the applicability of the previous result of ainsworth and vejchodsk 2011 in three aspects i arbitrary dimension ii mixed boundary conditions and iii non constant reaction coefficient it is the first robust upper bound on the error with these properties an auxiliary result that is of independent interest is the derivation of new explicit constants for two types of trace inequalities on simplices

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

computations of two fluid models based on a simple and robust hybrid primitive variable riemann solver with ausmd this paper is to continue our previous work in 2008 on solving a two fluid model for compressible liquid gas flows we proposed a pressure velocity based diffusion term original derived from ausmd scheme of wada and liou in 1997 to enhance its robustness the proposed ausmd schemes have been applied to gas and liquid fluids universally to capture fluid discontinuities such as the fluid interfaces and shock waves accurately for the ransom s faucet problem air water shock tube problems and 2d shock water liquid interaction problems however the proposed scheme failed at computing liquid gas interfaces in problems under large ratios of pressure density and volume of fraction the numerical instability has been remedied by chang and liou in 2007 using the exact riemann solver to enhance the accuracy and stability of numerical flux across the liquid gas interface here instead of the exact riemann solver we propose a simple ausmd type primitive variable riemann solver pvrs which can successfully solve 1d stiffened water air shock tube and 2d shock gas interaction problems under large ratios of pressure density and volume of fraction without the expensive cost of tedious computer time in addition the proposed approach is shown to deliver a good resolution of the shock front rarefaction and cavitation inside the evolution of high speed droplet impact on the wall

robust equilibrated residual error estimator for diffusion problems conforming elements this paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem the estimator which is a modification of those in d braess and j schoberl math comput 77 2008 pp 651 672 r verfurth siam j numer anal 47 2009 pp 3180 3194 is based on the prager synge identity and on a local recovery of an equilibrated flux numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator when the distribution of diffusion coefficients is local quasi monotone it is shown theoretically that the estimator is robust with respect to the size of jumps

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

estimating and localizing the algebraic and total numerical errors using flux reconstructions this paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization and an arbitrary iterative algebraic solver the derived bounds are based on the flux reconstruction techniques do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain we also discuss bounds on the discretization error their application for constructing mathematically justified stopping criteria for iterative algebraic solvers global and local efficiency of the total error upper bound and the relationship to the previously published estimates on the algebraic error theoretical results are illustrated on numerical experiments for higher order finite element approximations and the preconditioned conjugate gradient method they in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over thecomputational domain and illustrate the associate cost

a matched nonlinearity for phase estimation of a psk modulated carrier corresp a nonlinear phase estimator for a phase shift keyed psk modulated carrier has been developed by viterbi and viterbi their analysis is extended and an optimal or matched nonlinearity is derived for the estimator

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

how to solve compressible multifluid equations a simple robust and accurate method solving multifluid equations of compressible multiphase flows has proven to be extremely demanding because of some peculiar mathematical properties such as nonhyperbolicity nonconservative form and stiffness due to disparity in fluid properties and flow scales occurring typically in this paper we first consider the mathematical issues concerning nonhyperbolicity and nonconservative form their effects on the stability and convergence of numerical solutions are the theme of our presentation we shall present solutions for a range of problems selected to illuminate these numerical issues to this end we present a new numerical method that is simple to implement for a general class of fluids and yet is capable of robustly and accurately calculating phenomena involving material and shock discontinuities and interactions between them additionally the paper is completed with a new information for ensuring hyperbolicity under an interfacial pressure representation

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

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

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

a simple interface sharpening technique with a hyperbolic tangent function applied to compressible two fluid modeling a simple interface sharpening technique based on hyperbolic tangent interpolation which was proposed in the previous study f xiao y honma k kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer methods fluids 48 2005 1023 1040 is applied to the compressible two fluid modeling the implementation of this scheme is very simple the interpolation of the volume fraction in the monotonicity upwind scheme for conservation law muscl solver is just replaced by the hyperbolic tangent interpolation while the muscl interpolations for other variables are maintained this technique is limited for the region near the interface to prevent the spurious oscillations of a minor phase the one dimensional and two dimensional problems are solved and the results are compared with those of the original muscl solver the results show that the interface is significantly sharpened with this technique and its sharpness is well controlled by one parameter in addition the robustness of the scheme does not change with sharpening the interface in the range we investigated an interface sharpening technique is proposed for compressible two fluid modeling this method simply uses the hyperbolic tangent interpolation for volume fraction interface thickness is well controlled by one parameter in the scheme

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

two sided bounds for eigenvalues of differential operators with applications to friedrichs poincare trace and similar constants we present a general numerical method for computing guaranteed two sided bounds for principal eigenvalues of symmetric linear elliptic differential operators the approach is based on the galerkin method on the method of a priori a posteriori inequalities and on a complementarity technique the two sided bounds are formulated in a general hilbert space setting and as a byproduct we prove an abstract inequality of friedrichs poincare type the abstract results are then applied to friedrichs poincare and trace inequalities and fully computable two sided bounds on the optimal constants in these inequalities are obtained accuracy of the method is illustrated in numerical examples

a weakly compressible free surface flow solver for liquid gas systems using the volume of fluid approach this paper presents a weakly compressible volume of fluid formulation for modelling immiscible high density ratio two fluid flow under low mach number conditions this follows findings of experimental analyses that concluded the compressibility of the gas has a noteworthy effect on predicted pressure loads in liquid gas flow in certain instances with the aim of providing a more accurate numerical representation of dynamic two fluid flow the solver is subsequently extended to account for variations in gas densities a set of governing equations is proposed which accounts for the compressible properties of the gas phase in a manner which allows for a computationally efficient numerical simulation furthermore the governing equations are numerically expressed so that they allow for large variations in the material properties without introducing notable non physical oscillations over the interface for the discretisation of the governing equations an edge based vertex centred finite volume approach is followed the developed solver is applied to various test cases and demonstrated to be efficient and accurate

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

isogeometric methods for computational electromagnetics b spline and t spline discretizations in this paper we introduce methods for electromagnetic wave propagation based on splines and on t splines we define spline spaces which form a de rham complex and following the isogeometric paradigm we map them on domains which are piecewise spline or nurbs geometries we analyze their geometric and topological structure as related to the connectivity of the underlying mesh and we present degrees of freedom together with their physical interpretation the theory is then extended to the case of meshes with t junctions leveraging on the recent theory of t splines the use of t splines enhance our spline methods with local refinement capability and numerical tests show the efficiency and the accuracy of the techniques we propose

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

flux reconstructions in the lehmann goerisch method for lower bounds on eigenvalues abstract the standard application of the lehmann goerisch method for lower bounds on eigenvalues of symmetric elliptic second order partial differential operators relies on determination of fluxes i that approximate co gradients of exact eigenfunctions scaled by corresponding eigenvalues fluxes i are usually computed by solving a global saddle point problem with mixed finite element methods in this paper we propose a simpler global problem that yields fluxes i of the same quality the simplified problem is smaller it is positive definite and any h div conforming finite elements such as raviart thomas elements can be used for its solution in addition these global problems can be split into a number of independent local problems on patches which allows for trivial parallelization the computational performance of these approaches is illustrated by numerical examples for laplace and steklov type eigenvalue problems these examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions

a new flux splitting scheme a new flux splitting scheme is proposed the scheme is remarkably simple and yet its accuracy rivals and in some cases surpasses that of roe s solver in the euler and navier stokes solutions carried out in this study the scheme is robust and converges as fast as the roe splitting we propose an appropriately defined cell face advection mach number using values from the two straddling cells via associated characteristic speeds this interface mach number is then used to determine the upwind extrapolation for the convective quantities accordingly the name of the scheme is coined as the advection upstream splitting method ausm we also introduce a new pressure splitting which is shown to behave successfully yielding much smoother results than other existing pressure splittings of particular interest is the supersonic blunt body problem in which the roe scheme gives anomalous solutions the ausm produces correct solutions without difficulty for a wide range of flow conditions as well as grids