imposing dirichlet boundary conditions in the extended finite element method this paper is devoted to the imposition of dirichlet type conditions within the extended finite element method x fem this method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces imposing neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh observed numerically in earlier papers on the contrary much less work has been devoted to dirichlet boundary conditions for the x fem or the limiting case of stiff boundary conditions in this paper we introduce a strategy to impose dirichlet boundary conditions while preserving the optimal rate of convergence the key aspect is the construction of the correct lagrange multiplier space on the boundary as an application we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding rtm process while avoiding remeshing the case of inner conditions is also discussed as well as two important practical cases material interfaces and phase transformation front capturing copyright 2006 john wiley sons ltd

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

a variational multiscale discontinuous galerkin formulation for both implicit and explicit dynamic modeling of interfacial fracture abstract a stabilized discontinuous galerkin method is presented for modeling interfacial debonding under dynamic conditions the method s distinctive feature is its treatment of the inelastic gap at the interface as an internal variable that is evolved through a traction separation constitutive model enabling initially perfect adhesion to be captured numerical stability is derived using variational multiscale ideas and comparisons are drawn with both the intrinsic cohesive zone method and hybrid discontinuous galerkin cohesive zone method to expose subtle differences the proposed formulation is applied to model the nonlinear response of a composite unit cell under impact loading both explicit and implicit time integration schemes provide similar accuracy and stability and the critical time step for explicit analyses is only slightly reduced below the value for the continuous galerkin mesh without an interface further studies of mesh refinement and adjustments to the initial elastic interface stiffness in the reference cohesive zone model reveal the improved accuracy of the variational multiscale discontinuous galerkin method on coarser grids

an embedded dirichlet formulation for 3d continua this paper presents a new approach for imposing dirichlet conditions weakly on non fitting finite element meshes such conditions also called embedded dirichlet conditions are typically but not exclusively encountered when prescribing dirichlet conditions in the context of the extended finite element method xfem the methods key idea is the use of an additional stress field as the constraining lagrange multiplier function the resulting mixed hybrid formulation is applicable to 1d 2d and 3d problems the method does not require stabilization for the lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level furthermore only non zero diagonal terms are present in the tangent stiffness which allows the straight forward application of state of the art iterative solvers like algebraic multigrid amg techniques within this paper the method is applied to the linear momentum equation of an elastic continuum and to the transient incompressible navier stokes equations steady and unsteady benchmark computations show excellent agreement with reference values the general formulation presented in this paper can also be applied to other continuous field problems

a nurbs based hybrid collocation galerkin method for the analysis of boundary represented solids abstract the paper is concerned with a new numerical method nurbs based hybrid collocation galerkin method nurbs hcgm to solve the in plane motion problem of elastic solids it combines the merits of the so called scaled boundary finite element method sb fem and the isogeometric collocation method for the analysis the boundary scaling technique of sb fem is adopted it leads to a formulation where only the boundary of a structure is discretized here the nurbs basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary this is in accordance with the boundary representation modeling technique which is commonly employed in computer aided design software the inner domain is described by a radial scaling parameter applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation ode of euler type where the unknown displacements are a function of the radial scaling parameter in the present work a nurbs based collocation scheme is introduced to solve this equation nurbs basis functions are suggested for the approximation of the displacements in scaling direction the higher continuity provided by nurbs allows to use collocation to solve the ode directly instead of using the weak form in scaling direction the proposed approach is validated by comparison with the eigenvalue solution of the ode it is remarked that the eigenvalue solution is restricted to linear problems whereas the proposed method could be extended to nonlinear problems in general the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries the accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions the computational cost is investigated with the help of numerical examples and is compared to isogeometric galerkin approach

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

blending in the extended finite element method by discontinuous galerkin and assumed strain methods summary in the extended finite element method xfem errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain a discontinuous galerkin dg formulation is developed which circumvents this source of error a patch based version of the dg formulation is developed which decomposes the domain into enriched and unenriched subdomains continuity between patches is enforced with an internal penalty method an element based form is also developed where each element is considered a patch the patch based dg is shown to have similar accuracy to the element based dg for a given discretization but requires significantly fewer degrees of freedom the method is applied to material interfaces cracks and dislocation problems for the dislocations a contour integral form of the internal forces that only requires integration over the patch boundaries is developed a previously developed assumed strain as method is also developed further and compared with the dg method for weak discontinuities and linear elastic cracks the dg method is shown to be significantly more accurate than the standard xfem for a given element size and to converge optimally even where the standard xfem does not the accuracy of the dg method is similar to that of the as method but requires less application specific coding copyright q 2007 john wiley sons ltd

a bubble stabilized finite element method for dirichlet constraints on embedded interfaces we examine a bubble stabilized finite element method for enforcing dirichlet constraints on embedded interfaces by embedded we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh as such the robust imposition of dirichlet constraints using a lagrange multiplier field is not trivial to focus issues we consider a simple one sided problem that is representative of a wide class of evolving interface problems the bulk field is decomposed into coarse and fine scales giving rise to coarse scale and fine scale one sided sub problems the fine scale solution is approximated with bubble functions permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method importantly the method is simple to implement readily extends to multiple dimensions obviates the need to specify any free stabilization parameters and can lead to a symmetric positive definite system of equations the performance of the method is then examined through several numerical examples the accuracy of the lagrange multiplier is compared to results obtained using a local version of the domain integral method the variational multiscale approach proposed herein is shown to both stabilize the lagrange multiplier and improve the accuracy of the post processed fluxes copyright 2006 john wiley sons ltd

robust imposition of dirichlet boundary conditions on embedded surfaces summary we develop both stable and stabilized methods for imposing dirichlet constraints on embedded three dimensional surfaces in finite elements the stable method makes use of the vital vertex algorithm of bechet et al bechet et al 2009 int j numer meth engng 78 8 931 954 albeit with a modified set of discontinuous basis functions for the lagrange multiplier field on the embedded surface the stabilized method extends the work of dolbow and harari dolbow and harari 2009 int j numer meth engng 78 2 229 252 to three dimensional surfaces algorithmic and implementational details of both methods are provided several three dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches the results indicate that both methods yield optimal rates of convergence in various quantities of interest with the primary differences being in the surface flux the utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques published by john wiley sons ltd copyright c 2000 john wiley sons ltd

stable imposition of stiff constraints in explicit dynamics for embedded finite element methods summary we investigate various strategies to enforce the kinematics at an embedded interface for transient problems within the extended finite element method in particular we focus on explicit time integration of the semi discrete equations of motion and extend both dual and primal variational frameworks for constraint enforcement to a transient regime we reiterate the incompatibility of the dual formulation with purely explicit time integration and the severe restrictions placed by the courant friedrichs levy condition on primal formulations we propose an alternate consistent formulation for the primal method and derive an estimate for the stabilization parameter which is more amenable in an explicit dynamics framework importantly the use of the new estimate circumvents the need for any tolerances as an interface approaches an element boundary we also show that with interfacial constraints existing mass lumping schemes can lead to prohibitively small critical time steps accordingly we propose a mass lumping procedure which provides a more favorable estimate these techniques are then demonstrated on several benchmark numerical examples where we compare and contrast the accuracy of the primal methods against the dual methods in enforcing the constraints copyright 2012 john wiley sons ltd

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

on methods for stabilizing constraints over enriched interfaces in elasticity summary enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non smooth problems internal features such as boundaries are represented in such methods by using discontinuous enrichment of the standard finite element basis within such frameworks however imposition of interface constraints and or constitutive relations can cause unexpected difficulties depending upon how relevant fields are interpolated on un gridded interfaces this work address the stabilized treatment of constraints in an enriched finite element context both the lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields we demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner in a bubble stabilized approach fine scale degrees of freedom are added over elements supporting the interface the variational form can be shown to have a similar form to a second approach we consider nitsche s method with the exception that the stabilization terms follow directly from the bubble functions in this work we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two dimensional elasticity we examine several benchmark problems in elasticity and show that only nitsche s method and the bubble stabilization approach produce stable traction fields over internal boundaries we also demonstrate a novel difference between the penalty method and nitsche s method in that the latter passes the patch test exactly regardless of the stabilization parameter s magnitude results for more complicated geometries and triple interface junctions are also presented copyright 2008 john wiley sons ltd

imposing dirichlet boundary conditions in hierarchical cartesian meshes by means of stabilized lagrange multipliers this is the pre peer reviewed version of the following article tur m albelda j nadal e and rodenas j j 2014 imposing dirichlet boundary conditions in hierarchical cartesian meshes by means of stabilized lagrange multipliers int j numer meth engng 98 399 417 which has been published in final form at http dx doi org 10 1002 nme 4629 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

nitsche s method method for mixed dimensional analysis conforming and non conforming continuum beam and continuum plate coupling a nitche s method is presented to couple dierent mechanical models they include coupling of a solid and a beam and of a solid and a plate both conforming and non conforming formulations are presented in a non conforming formulation the structure domain is overlapped by a rened solid model applications can be found in multi dimensional analyses in which parts of a structure are modeled with solid elements and others are modeled using a coarser model with beam and or plate elements discretisations are performed using both standard lagrange elements and high order nurbs non uniform rational bsplines based isogeometric elements we present various examples to demonstrate the performance of the method

weak coupling for isogeometric analysis of non matching and trimmed multi patch geometries nitsche s method can be used as a coupling tool for non matching discretizations by weakly enforcing interface constraints we explore the use of weak coupling based on nitsche s method in the context of higher order and higher continuity b splines and nurbs we demonstrate that weakly coupled spline discretizations do not compromise the accuracy of isogeometric analysis we show that the combination of weak coupling with the finite cell method opens the door for a truly isogeometric treatment of trimmed b spline and nurbs geometries that eliminates the need for costly reparameterization procedures we test our methodology for several relevant technical problems in two and three dimensions such as gluing together trimmed multi patches and connecting non matching meshes that contain b spline basis functions and standard triangular finite elements the results demonstrate that the concept of nitsche based weak coupling in conjunction with the finite cell method has the potential to considerably increase the flexibility of the design through analysis process in isogeometric analysis

remark on a relation between magnetic field and topological degree of order parameter in a type ii superconductivity model abstract it is proved that in the absence of an applied magnetic field the two space dimensional steady state helmholtz free energy which models the type ii superconductors admits no minimizer in the following admissible function set u a h 1 c h 1 r 2 u 1 1 2 i u d u ud u d 2k here c is the complex plane k is a fixed nonzero integer which is the topological degree of u with respect to its value 0 the case k 0 is discussed separately under different assumptions

a nitsche type formulation and comparison of the most common domain decomposition methods in isogeometric analysis this paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis the methods comprise a penalty approach lagrange multiplier methods and a nitsche type method for the nitsche method a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity all these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2d in particular focus is put on non uniform rational b spline patches which join nonconformingly along their common interface thus the application of isogeometric analysis is extended to multi patches which can have an arbitrary parametrization on the adjacent edges moreover it has been shown that the unique properties provided by isogeometric analysis that is high order functions and smoothness across the element boundaries carry over for the analysis of multiple domains copyright 2013 john wiley sons ltd

fast online graph clustering via erdős renyi mixture in the context of graph clustering we consider the problem of simultaneously estimating both the partition of the graph nodes and the parameters of an underlying mixture of affiliation networks in numerous applications the rapid increase of data size over time makes classical clustering algorithms too slow because of the high computational cost in such situations online clustering algorithms are an efficient alternative to classical batch algorithms we present an original online algorithm for graph clustering based on a erdos renyi graph mixture the relevance of the algorithm is illustrated using both simulated and real data sets the real data set is a network extracted from the french political blogosphere and presents an interesting community organization

an extended finite element method with algebraic constraints xfem ac for problems with weak discontinuities abstract we present a new extended finite element method with algebraic constraints xfem ac for recovering weakly discontinuous solutions across internal element interfaces if necessary cut elements are further partitioned by a local secondary cut into body fitting subelements each resulting subelement contributes an enrichment of the parent element the enriched solutions are then tied using algebraic constraints which enforce c 0 continuity across both cuts these constraints impose equivalence of the enriched and body fitted finite element solutions and are the key differentiating feature of the xfem ac in so doing a stable mixed formulation is obtained without having to explicitly construct a compatible lagrange multiplier space and prove a formal inf sup condition likewise convergence of the xfem ac solution follows from its equivalence to the interface fitted finite element solution this relationship is further exploited to improve the numerical solution of the resulting xfem ac linear system examples are shown demonstrating the new approach for both steady state and transient diffusion problems

a scaled boundary isogeometric formulation for the elasto plastic analysis of solids in boundary representation abstract this contribution deals with the nonlinear analysis of boundary represented solids with elasto plastic material behavior based on the so called scaled boundary isogeometric formulation sb iga the proposed approach combines the features of the scaled boundary finite element method and isogeometric analysis based on the original boundary representation of the cad model a formulation is provided where the geometrical description of the boundary is sufficient to define the entire surface the domain is parameterized by a radial scaling parameter emanating from a scaling center and a parameter in circumferential direction along the boundary non star shaped domains are tackled by standard sub structuring here conforming discretizations are considered for the two dimensional case according to the isogeometric paradigm nurbs basis functions are employed for the approximation of the solution the displacement response is derived based on a multiplicative decomposition of the approximation in circumferential and radial scaling direction the boundary value problem is solved with the galerkin method the newton raphson iterative scheme is employed to obtain the nonlinear response several benchmark tests demonstrate the accuracy and computational efficiency of the formulation

a robust nitsche s formulation for interface problems with spline based finite elements summary the extended finite element method x fem has proven to be an accurate robust method for solving embedded interface problems with a few exceptions the x fem has mostly been used in conjunction with piecewise linear shape functions and an associated piecewise linear geometrical representation of interfaces in the current work the use of spline based finite elements is examined along with a nitsche technique for enforcing constraints on an embedded interface to obtain optimal rates of convergence we employ a hierarchical local refinement approach to improve the geometrical representation of curved interfaces we further propose a novel weighting for the interfacial consistency terms arising in the nitsche variational form with b splines a qualitative dependence between the weights and the stabilization parameters is established with additional element level eigenvalue calculations an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of large heterogeneities as well as elements with arbitrarily small volume fractions we demonstrate the accuracy and robustness of the proposed method through several numerical examples copyright 2015 john wiley sons ltd

a new formulation for imposing dirichlet boundary conditions on non matching meshes generating matching meshes for problems with complex boundaries is often an intricate process and the use of non matching meshes appears as an appealing solution yet enforcing boundary conditions on non matching meshes is not a straightforward process especially when prescribing those of dirichlet type by combining a type of generalized finite element method gfem with the lagrange multiplier method a new technique for the treatment of essential boundary conditions on non matching meshes is introduced in this manuscript the new formulation yields a symmetric stiffness matrix and is straightforward to implement as a result the methodology makes possible the analysis of problems with the use of simple structured meshes irrespective of the problem domain boundary through the solution of linear elastic problems we show that the optimal rate of convergence is preserved for piecewise linear finite elements yet the formulation is general and thus it can be extended to other elliptic boundary value problems

primal interface formulation for coupling multiple pdes a consistent derivation via the variational multiscale method abstract this paper presents a primal interface formulation that is derived in a systematic manner from a lagrange multiplier method to provide a consistent framework to couple different partial differential equations pde as well as to tie together nonconforming meshes the derivation relies crucially on concepts from the variational multiscale vms approach wherein an additive multiscale decomposition is applied to the primary solution field modeling the fine scales locally at the interface using bubble functions consistent residual based terms on the boundary are obtained that are subsequently embedded into the coarse scale problem the resulting stabilized lagrange multiplier formulation is converted into a robust discontinuous galerkin dg method by employing a discontinuous interpolation of the multipliers along the segments of the interface as a byproduct analytical expressions are derived for the stabilizing terms and weighted numerical flux that reflect the jump in material properties governing equation or element geometry across the interface also a procedure is proposed for automatically generating the fine scale bubble functions that is inspired by a performance study of residual free bubbles for the interface problem a series of numerical tests confirms the robustness of the method for solving interface problems with heterogeneous elements materials and or governing equations and also highlights the benefit and importance of deriving the flux and stabilization terms

the nitsche method applied to a class of mixed dimensional coupling problems abstract a computational approach for the mixed dimensional modeling of time harmonic waves in elastic structures is proposed a two dimensional 2d structure is considered that includes a part which is assumed to behave in a one dimensional 1d way the 2d and 1d structural regions are discretized using 2d and 1d finite element formulations the coupling of the 2d and 1d regions is performed weakly by using the nitsche method the advantage of using the nitsche method to impose boundary and interface conditions has been demonstrated by various authors here this advantage is shown in the context of mixed dimensional coupling the computational aspects of the method are discussed and it is compared to the slightly simpler penalty method both theoretically and numerically numerical examples are presented in various configurations where the 1d model is either confined laterally or laterally free and where the 2d part is either simply connected or doubly connected the performance is investigated for various wave numbers and various extents of the 1d region varying material properties and distributed loads in the 1d and 2d parts are also considered it is concluded that the nitsche method is a viable technique for mixed dimensional coupling of elliptic problems of this type

stabilized low order finite elements for frictional contact with the extended finite element method abstract contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity in which the normal contact pressure exhibits spurious oscillation this oscillation does not go away with mesh refinement and in some cases it even gets worse as the mesh is refined using a lagrange multipliers formulation we trace this problem to non satisfaction of the lbb condition associated with equal order interpolation of slip and normal component of traction in this paper we employ a stabilized finite element formulation based on the polynomial pressure projection ppp technique which was used successfully for stokes equation and for coupled solid deformation fluid diffusion using low order mixed finite elements for the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method we use low order linear triangular elements tetrahedral elements for 3d for both slip and normal pressure degrees of freedom and show the efficacy of the stabilized formulation on a variety of plane strain plane stress and three dimensional problems

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

fluid structure interaction by the discontinuous galerkin method for large deformations a discontinuous galerkin method for large deformation fluid structure interaction problems is developed the fluid structure interface can be arbitrarily aligned relative to the fluid grid an eulerian description is used for the fluid with a lagrangian description of the solid results are presented for several examples that show excellent agreement with immersed fictitious element methods copyright 2008 john wiley sons ltd

a study on unfitted 1d finite element methods in the present paper we consider a 1d poisson model characterized by the presence of an interface where a transmission condition arises due to jumps of the coefficients we aim at studying finite element methods with meshes not fitting such an interface it is well known that when the mesh does not fit the material discontinuities the resulting scheme provides in general lower order accurate solutions we focus on so called embedded approaches frequently adopted to treat fluid structure interaction problems with the aim of recovering higher order of approximation also in presence of non fitting meshes we implement several methods inspired by the immersed boundary method the fictitious domain method and the extended finite element method in particular we present four formulations in a comprehensive and unified format proposing several numerical tests and discussing their performance moreover we point out issues that may be encountered in the generalization to higher dimensions and we comment on possible solutions

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd