error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

a priori and a posteriori error analyses of a high order unfitted mixed fem for stokes flow abstract we propose and analyze a high order unfitted mixed finite element method for the pseudostress velocity formulation of the stokes problem with dirichlet boundary condition on a fluid domain with curved boundary the method consists of approximating by a polyhedral subdomain d h with boundary h where a galerkin method is applied to approximate the solution and on a transferring technique based on integrating the extrapolated discrete gradient of the velocity to approximate the dirichlet boundary data on the computational boundary h the associated galerkin scheme is defined by raviart thomas elements of order k 0 for the pseudostress and discontinuous polynomials of degree k for the velocity provided suitable hypotheses on the mesh near the boundary we prove well posedness of the galerkin scheme by means of the babuska brezzi theory and establish the corresponding optimal convergence of o h k 1 moreover for the case when h is constructed through a piecewise linear interpolation of we propose a reliable and quasi efficient residual based a posteriori error estimator its definition makes use of a postprocessed velocity with enhanced accuracy to achieve the same rate of convergence of the method when the solution is smooth enough numerical experiments illustrate the performance of the scheme show the behavior of the associated adaptive algorithm and validate the theory

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

a residual based a posteriori error estimator for a fully mixed formulation of the stokes darcy coupled problem abstract in this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2d flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we consider dual mixed formulations in both media which yields the pseudostress and the velocity in the fluid together with the velocity and the pressure in the porous medium and the traces of the porous media pressure and the fluid velocity on the interface as the resulting unknowns the set of feasible finite element subspaces includes raviart thomas elements of lowest order and piecewise constants for the velocities and pressures respectively in both domains together with continuous piecewise linear elements for the traces we derive a reliable and efficient residual based a posteriori error estimator for the coupled problem the proof of reliability makes use of the global inf sup condition helmholtz decompositions in both media and local approximation properties of the clement interpolant and raviart thomas operator on the other hand inverse inequalities the localization technique based on element bubble and edge bubble functions and known results from previous works are the main tools for proving the efficiency of the estimator finally some numerical results confirming the theoretical properties of this estimator and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution are reported

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 projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

the shifted boundary method for embedded domain computations part i poisson and stokes problems abstract we propose a new finite element method for embedded domain computations which falls in the category of surrogate approximate boundary algorithms the key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary and to appropriately modify the shifted boundary conditions enforced weakly in order to preserve optimal convergence rates of the numerical solution this process yields a method which in our view is simple efficient and also robust since it is not affected by the small cut cell problem although general in nature here we apply this new concept to the poisson and stokes problems we present in particular the full analysis of stability and convergence for the case of the poisson operator and numerical tests for both the poisson and stokes equations for geometries of progressively higher complexity

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

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 priori error analysis for hdg methods using extensions from subdomains to achieve boundary conformity we present the first a priori error analysis of a technique that allows us to numerically solve steady state diffusion problems defined on curved domains by using finite element methods defined in polyhedral subdomains dh for a wide variety of hybridizable discontinuous galerkin and mixed methods we prove that the order of convergence in the l2 norm of the approximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain and that of the computational domain h is of order h we also prove that the l 2 norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between and h is of order h 5 4 but with only half an additional order when such a distance is of order h finally we present numerical experiments confirming the theoretical results and showing that even when the distance between and h is of order h the above mentioned projection of the error of the scalar variable can still superconverge with a full additional order

an a posteriori error estimate for the variable degree raviart thomas method we propose a new a posteriori error analysis of the variable degree hybridized version of the raviart thomas method for second order elliptic problems on conforming meshes made of simplexes we establish both the reliability and eciency of the estimator for the l2 norm of the error of the ux we also nd the explicit dependence of the estimator on the order of the local spaces k 0 the only constants that are not explicitly computed are those depending on the shape regularity of the simplexes in particular the constant of the local eciency in equality is proven to behave like k 2 3 2 however we present numer ical experiments suggesting that such constant is actually independent of k

coupling at a distance hdg and bem we introduce a new technique for numerically solving exterior dirichlet boundary value problems for second order elliptic equations it consists of coupling a hybridizable discontinuous galerkin hdg method used for solving the so called interior problem on a bounded region containing the support of the source term with a boundary element method bem for solving the problem exterior to that region the novelty is that the bem is defined on a suitably chosen smooth artificial boundary whereas the hdg method is defined on a polyhedral subdomain because of the choice of the artificial boundary we can take advantage of the spectral convergence of the bem solution and of the simplicity of the corresponding equations because the hdg method is defined on a polyhedral subdomain there is no need to try to fit the mesh to the artificial boundary instead the hdg is coupled at a distance with the bem by using simple dirichlet to neumann operators defined in the region between the artificial boundary and the polyhedral subdomain numerical experiments displaying the performance of the new technique are presented optimal orders of convergence are obtained even though the distance between the artificial boundary and the polyhedral domain is of order h

fictitious domain finite element methods using cut elements i a stabilized lagrange multiplier method we propose a fictitious domain method where the mesh is cut by the boundary the primal solution is computed only up to the boundary the solution itself is defined also by nodes outside the domain

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

the shifted boundary method for embedded domain computations part ii linear advection diffusion and incompressible navier stokes equations abstract we propose a new embedded finite element method for the linear advection diffusion equation and the laminar and turbulent incompressible navier stokes equations the proposed method belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates we include the full analysis of stability and convergence of the method in the linear advection diffusion equation and a battery of tests for the case of laminar and turbulent incompressible navier stokes equations we also discuss the conservation properties of the proposed method in all cases

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

a posteriori error estimates for a mixed fem formulation of a non linear elliptic problem abstract we consider the numerical solution via the mixed finite element method of a non linear elliptic partial differential equation in divergence form with dirichlet boundary conditions besides the temperature u and the flux we introduce u as a further unknown which yields a variational formulation with a twofold saddle point structure we derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem which does not need any equilibrium property for its solvability in addition for specific finite element subspaces of raviart thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems our approach does not need the exact finite element solution but any reasonable approximation of it such as for instance the one obtained with a fully discrete galerkin scheme in particular we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi linear forms involved finally several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions

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

on the construction of infinitely many congruence classes of imbedded closed minimal hypersurfaces in s n 1 for all n geq 3

on the completion of qualitative possibility measures a relational description of a possibility introduced by d dubois and h prade in their 1988 book called a qualitative possibility measure is discussed among the requirements of these kinds of relationships is that they be complete the focus is on the issue of completing these relationships a measure associated with weak orderings called buoyancy is introduced the principle of maximal buoyancy is suggested as a means for completion the connection between the principle of maximal buoyancy and the principle of minimal specificity is shown a special family of buoyancy measures called badd buoyancies is also introduced that have the properties of entropy measures when the aggregates are restricted to be a probability distribution

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

mixed finite element methods for linear elasticity with weakly imposed symmetry in this paper we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima tions to both stresses and displacements the methods are based on a modified form of the hellinger reissner variational principle that only weakly imposes the symmetry condition on the stresses although this approach has been previously used by a number of authors a key new ingredient here is a constructive derivation of the elasticity complex starting from the de rham complex by mimicking this construction in the discrete case we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de rham complex these elements appear to be simpler than the ones previously derived for example we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field

solving dirichlet boundary value problems on curved domains by extensions from subdomains we present a technique for numerically solving dirichlet boundary value problems for second order elliptic equations on domains omega with curved boundaries this is achieved by using suitably defined extensions from polyhedral subdomains textsf d h the problem of dealing with curved boundaries is thus reduced to the evaluations of simple line integrals the technique is independent of the representation of the boundary of the space dimension and allows for the use of only polyhedral elements we apply this technique to the hybridizable discontinuous galerkin method and provide extensive numerical experiments showing that the convergence properties of the resulting method are the same as those for the case in which omega textsf d h whenever the distance of textsf d h to partial omega is of order h in particular we find that when using polynomial approximations of degree k on the polyhedral subdomains textsf d h both the scalar and vector unknowns converge with the optimal order of k 1 in the whole domain omega for k ge0 moreover we also find that for k ge1 both a postprocessing of the scalar unknown in the polyhedral subdomain textsf d h as well as its extension from it converge with order k 2 the extension also converges with order 2 for k 0

a high order discontinuous galerkin method for compressible flows with immersed boundaries summary we present a higher order discretization scheme for the compressible euler and navier stokes equations with immersed boundaries our approach makes use of a discontinuous galerkin dg discretization in a domain that is implicitly defined by means of a level set function the zero iso contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary fitted cells in order to retain the full order of convergence of the scheme it is crucial to perform volume and surface integrals in boundary cells with high accuracy this is achieved using a linear moment fitting strategy moreover we apply a non intrusive cell agglomeration technique that averts problems with very small and ill shaped cuts the robustness accuracy and convergence properties of the scheme are assessed in several two dimensional test cases for the steady compressible euler and navier stokes equations approximation orders range from zero to four even though the approach directly generalizes to even higher orders in all test cases with a sufficiently smooth solution the experimental order of convergence matches the expected rate for dg schemes this article is protected by copyright all rights reserved

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