a hybridizable discontinuous galerkin method for stokes flow abstract in this paper we introduce a hybridizable discontinuous galerkin method for stokes flow the method is devised by using the discontinuous galerkin methodology to discretize a velocity pressure gradient formulation of the stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization one of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries thereby leading to a significant reduction in the size of the resulting matrix moreover by using an augmented lagrangian method the globally coupled unknowns are further reduced to the numerical trace of the velocity only another important feature is that the approximations of the velocity pressure and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the approximate variables based on the optimal convergence of the hdg method we apply an element by element postprocessing scheme to obtain a new approximate velocity which converges with order k 2 in the l 2 norm for k 1 the postprocessing performed at the element level is less expensive than the solution procedure numerical results are provided to assess the performance of the method

collision in a cross shaped domain a steady 2d navier stokes example demonstrating the importance of mass conservation in cfd abstract in the numerical simulation of the incompressible navier stokes equations different numerical instabilities can occur while instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete ladyzhenskaya babuska brezzi lbb constant are well known instability in the discrete velocity due to a poor mass conservation at high reynolds numbers sometimes seems to be underestimated at least when using conforming galerkin mixed finite element methods like the taylor hood element the classical grad div stabilization for enhancing discrete mass conservation is often neglected in practical computations though simple academic flow problems showing the importance of mass conservation are well known these examples differ from practically relevant ones since specially designed force vectors are prescribed therefore we present a simple steady navier stokes problem in two space dimensions at reynolds number 1024 a colliding flow in a cross shaped domain where the instability of poor mass conservation is studied in detail and where no force vector is prescribed

principal minors and diagonal similarity of matrices abstract let a b be n n matrices with entries in a field f our purpose is to show the following theorem suppose n 4 a is irreducible and for every partition of 1 2 n into subsets with 2 2 either rank a 2 or rank a 2 if a and b have equal corresponding principal minors of all orders then b or b t is diagonally similar to a

on really locking free mixed finite element methods for the transient incompressible stokes equations inf sup stable mixed methods for the steady incompressible stokes equations that relax the divergence constraint are often claimed to deliver locking free discretizations however this relaxation leads to a pressure dependent contribution in the velocity error which is proportional to the inverse of the viscosity thus giving rise to a different locking phenomenon however a recently proposed modification of the right hand side alone leads to a discretization that is really locking free i e its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity in this contribution we extend this approach to the transient incompressible stokes equations where besides the right hand side also the velocity time derivative requires an improved space discretization semidiscrete and fully discrete a priori velocity and pressure error estimates are derived which show remarkable robustness properties two numerical examples illustrate the superior accura

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

a hybrid high order method for the incompressible navier stokes problem robust for large irrotational body forces abstract we develop a novel hybrid high order method for the incompressible navier stokes problem robust for large irrotational body forces the key ingredients of the method are discrete versions of the body force and convective contributions in the momentum equation formulated in terms of a globally divergence free velocity reconstruction two key properties are mimicked at the discrete level namely the invariance of the velocity with respect to irrotational body forces and the non dissipativity of the convective term a full convergence analysis is carried out showing optimal orders of convergence under a smallness condition involving only the solenoidal part of the body force the performance of the method is illustrated by a complete panel of numerical tests including comparisons that highlight the benefits with respect to more standard formulations

on the divergence constraint in mixed finite element methods for incompressible flows the divergence constraint of the incompressible navier stokes equations is revisited in the mixed finite element framework while many stable and convergent mixed elements have been developed throughout the past four decades most classical methods relax the divergence constraint and only enforce the condition discretely as a result these methods introduce a pressure dependent consistency error which can potentially pollute the computed velocity these methods are not robust in the sense that a contribution from the right hand side which influences only the pressure in the continuous equations impacts both velocity and pressure in the discrete equations this article reviews the theory and practical implications of relaxing the divergence constraint several approaches for improving the discrete mass balance or even for computing divergence free solutions will be discussed grad div stabilization higher order mixed methods derived on the basis of an exact de rham complex h div conforming finite

on the role of the helmholtz decomposition in mixed methods for incompressible flows and a new variational crime abstract according to the helmholtz decomposition the irrotational parts of the momentum balance equations of the incompressible navier stokes equations are balanced by the pressure gradient unfortunately nearly all mixed methods for incompressible flows violate this fundamental property resulting in the well known numerical instability of poor mass conservation the origin of this problem is the lack of l 2 orthogonality between discretely divergence free velocities and irrotational vector fields therefore a new variational crime for the nonconforming crouzeix raviart element is proposed where divergence free lowest order raviart thomas velocity reconstructions reestablish l 2 orthogonality this approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence free flow solvers in the stokes case optimal a priori error estimates for the velocity gradients and the pressure are derived moreover the discrete velocity is independent of the continuous pressure several detailed linear and nonlinear numerical examples illustrate the theoretical findings

a hybridized finite element method for the stokes problem a locally conservative hybridized finite element method for stokes equations is presented and analyzed the hybridized approach reduces a lot of degrees of freedom especially for pressure approximation in our approach the pressure is determined locally up to a constant therefore the global stiffness system contains only the average of pressure variable on each cell as unknowns

energy stable and momentum conserving hybrid finite element method for the incompressible navier stokes equations a hybrid method for the incompressible navier stokes equations is presented the method inherits the attractive stabilizing mechanism of upwinded discontinuous galerkin methods when momentum advection becomes significant equal order interpolations can be used for the velocity and pressure fields and mass can be conserved locally using continuous lagrange multiplier spaces to enforce flux continuity across cell facets the number of global degrees of freedom is the same as for a continuous galerkin method on the same mesh different from our earlier investigations on the approach for the navier stokes equations the pressure field in this work is discontinuous across cell boundaries it is shown that this leads to very good local mass conservation and for an appropriate choice of finite element spaces momentum conservation also a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed even in the absence of a pointwise solenoidal velocity field mass conservation momentum conservation and global energy stability are proved for the time continuous case and for a fully discrete scheme the presented analysis results are supported by a range of numerical simulations

on velocity errors due to irrotational forces in the navier stokes momentum balance this contribution studies the influence of the pressure on the velocity error in finite element discretisations of the navier stokes equations four simple benchmark problems that are all close to real world applications convey that the pressure can be comparably large and is not to be underestimated in fact the velocity error can be arbitrarily large in such situations only pressure robust mixed finite element methods whose velocity error is pressure independent can avoid this influence indeed the presented examples show that the pressure dependent component in velocity error estimates for classical mixed finite element methods is sharp in consequence classical mixed finite element methods are not able to simulate some classes of real world flows even in cases where dominant convection and turbulence do not play a role

on lossless feedback delay networks lossless feedback delay networks fdns are commonly used as a design prototype for artificial reverberation algorithms the lossless property is dependent on the feedback matrix which connects the output of a set of delays to their inputs and the lengths of the delays both unitary and triangular feedback matrices are known to constitute lossless fdns however the most general class of lossless feedback matrices has not been identified in this contribution it is shown that the fdn is lossless for any set of delays if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix the necessity of the generalized class of feedback matrices is demonstrated by examples of fdn designs proposed in literature

new locally conservative finite element methods on a rectangular mesh a new family of locally conservative finite element methods for a rectangular mesh is introduced to solve second order elliptic equations our approach is composed of generating pde adapted local basis and solving a global matrix system arising from a flux continuity equation quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations next this approach is exploited to approximate stokes equations numerical results are presented for various examples including the lid driven cavity problem

implementation of discontinuous skeletal methods on arbitrary dimensional polytopal meshes using generic programming discontinuous skeletal methods approximate the solution of boundary value problems by attaching discrete unknowns to mesh faces hence the term skeletal while allowing these discrete unknowns to be chosen independently on each mesh face hence the term discontinuous cell based unknowns which can be eliminated locally by a schur complement technique also known as static condensation are also used in the formulation salient examples of high order discontinuous skeletal methods are hybridizable discontinuous galerkin methods and the recently devised hybrid high order methods some major benefits of discontinuous skeletal methods are that their construction is dimension independent and that they offer the possibility to use general meshes with polytopal cells and non matching interfaces in this work we show how this mathematical flexibility can be efficiently replicated in a numerical software using generic programming we describe a number of generic algorithms and data structures for high order discontinuous skeletal methods within a write once run on any kind of mesh framework the computational efficiency of the implementation is assessed on the poisson model problem discretized using various polytopal meshes and the hybrid high order method

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

diagonally scaled permutations and circulant matrices abstract if r is an n n matrix over the complex field which is the product of a diagonal matrix d and a permutation matrix p then r is called a diagonally scaled permutation matrix we present the eigenstructure of r by observing that r is permutation similar to the direct sum of diagonally scaled permutation matrices of the form dc where d is a diagonal matrix and c is the circulant permutation 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 the matrix dc is called a scaled circulant permutation matrix we consider two cases for r dc when the scaling matrix d is nonsingular and when d is singular in the singular case r is nilpotent and we are able to obtain upper and lower bounds on the index of nilpotency of r we conclude with information about matrices that commute with a scaled permutation matrix we are also able to represent an arbitrary n n toeplitz matrix as a sum of matrices of the form d k ck for k 1 n where d k is a diagonal matrix

a discontinuous skeletal method for the viscosity dependent stokes problem we devise and analyze arbitrary order nonconforming methods for the discretization of the viscosity dependent stokes equations on simplicial meshes we keep track explicitly of the viscosity and aim at pressure robust schemes that can deal with the practically relevant case of body forces with large curl free part in a way that the discrete velocity error is not spoiled by large pressures the method is inspired from the recent hybrid high order hho methods for linear elasticity after elimination of the auxiliary variables by static condensation the linear system to be solved involves only discrete face based velocities which are polynomials of degree k ge 0 and cell wise constant pressures our main result is a pressure independent energy error estimate on the velocity of order k 1 the main ingredient to achieve pressure independence is the use of a divergence preserving velocity reconstruction operator in the discretization of the body forces we also prove an l 2 pressure estimate of order k 1 and an l 2 velocity estimate of order k 2 the latter under elliptic regularity the local mass and momentum conservation properties of the discretization are also established finally two and three dimensional numerical results are presented to support the analysis

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

a hybrid high order method for finite elastoplastic deformations within a logarithmic strain framework we devise and evaluate numerically a hybrid high order hho method for finite plasticity within a logarithmic strain framework the hho method uses as discrete unknowns piecewise polynomials of order k 1 on the mesh skeleton together with cell based polynomi als that can be eliminated locally by static condensation the hho method leads to a primal formulation supports polyhedral meshes with non matching interfaces is free of volumetric locking the integration of the behavior law is performed only at cell based quadrature nodes and the tangent matrix in newton s method is symmetric moreover the principle of virtual work is satisfied locally with equilibrated tractions various two and three dimensional benchmarks are presented as well as comparison against known solutions with an industrial software using conforming and mixed finite elements

the derivation of hybridizable discontinuous galerkin methods for stokes flow in this paper we introduce a new class of discontinuous galerkin methods for the stokes equations the main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries we present four ways of hybridizing the methods which differ by the choice of the globally coupled unknowns classical methods for the stokes equations can be thought of as limiting cases of these new methods

a discontinuous skeletal method for advection diffusion reaction on general meshes we design and analyze an approximation method for advection diffusion reaction equa tions where the generalized degrees of freedom are polynomials of order k 0 at mesh faces the method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions fairly general meshes with poly topal and nonmatching cells are supported arbitrary polynomial orders can be considered including the case k 0 which is closely related to mimetic finite difference mixed hybrid finite volume methods the error analysis covers the full range of peclet numbers including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications numerical results are presented

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

an abstract analysis framework for monolithic discretisations of poroelasticity with application to hybrid high order methods abstract in this work we introduce a novel abstract framework for the stability and convergence analysis of fully coupled discretisations of the poroelasticity problem and apply it to the analysis of hybrid high order hho schemes a relevant feature of the proposed framework is that it rests on mild time regularity assumptions that can be derived from an appropriate weak formulation of the continuous problem to the best of our knowledge these regularity results for the biot problem are new a novel family of hho discretisation schemes is also proposed and analysed and their performance numerically evaluated

a hybrid high order method for a class of quasi newtonian stokes equations on general meshes abstract in this paper we introduce a hybrid high order hho discrete scheme for numerically solving a class of incompressible quasi newtonian stokes equations in r 2 the presented hho method depends on hybrid discrete velocity unknowns at cells and edges and pressure unknowns at cells benefiting from the hybridization of unknowns the computation cost can be reduced by the technique of static condensation and the solvability of the static condensation algebra system is proved furthermore we study the hho scheme by polynomials of arbitrary degrees k k 1 on the general meshes and geometries the unique solvability of the discrete scheme is proved additionally the optimal a priori error estimates for the velocity gradient and pressure approximations are obtained finally we provide several numerical results to verify the good performance of the proposed hho scheme and confirm the optimal approximation properties on a variety of meshes and geometries

stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection abstract we consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation free forcing in the momentum equation we provide analysis that suggests for such problems obtaining accurate solutions necessitates either the use of pointwise divergence free finite elements such as scott vogelius or heavy grad div stabilization of weakly divergence free elements the theory is demonstrated in numerical experiments for a benchmark natural convection problem where large irrotational forcing occurs with high rayleigh numbers

high order exactly divergence free hybrid discontinuous galerkin methods for unsteady incompressible flows abstract in this paper we present an efficient discretization method for the solution of the unsteady incompressible navier stokes equations based on a high order hybrid discontinuous galerkin formulation the crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non linear parts with respect to their temporal and spatial treatment exploiting the flexibility of operator splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub problems a corresponding hyperbolic transport problem and an unsteady stokes problem for the hyperbolic transport problem a spatial discretization with an upwind discontinuous galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation the treatment of the stokes part involves the solution of linear systems in this case a discretization with hybrid discontinuous galerkin methods is better suited we consider such a discretization for the stokes part with two important features h div conforming finite elements to guarantee exactly divergence free velocity solutions and a projection operator which reduces the number of globally coupled unknowns we present the method discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

a hybrid high order method for the incompressible navier stokes equations based on temam s device abstract in this work we propose a novel hybrid high order method for the incompressible navier stokes equations based on a formulation of the convective term including temam s device for stability the proposed method has several advantageous features it supports arbitrary approximation orders on general meshes including polyhedral elements and non matching interfaces it is inf sup stable it is locally conservative it supports both the weak and strong enforcement of velocity boundary conditions it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element particular care is devoted to the design of the convective trilinear form which mimicks at the discrete level the non dissipation property of the continuous one the possibility to add a convective stabilisation term is also contemplated and a formulation covering various classical options is discussed the proposed method is theoretically analysed and an energy error estimate in h k 1 with h denoting the meshsize is proved under the usual data smallness assumption a thorough numerical validation on two and three dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations including in particular the well known two and three dimensional lid driven cavity problems

a high order mimetic method on unstructured polyhedral meshes for the diffusion equation abstract we present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes these mimetic discretizations are built to satisfy local consistency and stability conditions the consistency condition is an exactness property i e the mimetic schemes are exact when the solution is a polynomial of an assigned degree the stability condition ensures the well posedness of the method the degrees of freedom are the solution moments on mesh faces and inside mesh cells higher order schemes are built using higher order moments the developed schemes are verified numerically on diffusion problems with constant and spatially variable possibly discontinuous tensorial coefficients

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

a modified weak galerkin finite element method for the stokes equations a modified weak galerkin mwg finite element method is introduced for the stokes equations in the primary velocity pressure formulation this method is based on the weak galerkin wg finite element method developed recently in wang and ye 0000 the unknowns associated with element boundaries in the wg method have been eliminated in the mwg method as the result the mwg method has fewer degrees of freedom optimal order error estimates are established for the corresponding numerical approximation in various norms numerical results are presented to confirm the theory

computational methods for continuum models of platelet aggregation platelet aggregation plays an important role in blood clotting robust numerical methods for simulating the behavior of fogelson s continuum models of platelet aggregation have been developed which in particular involve a hybrid finite difference and spectral method for the models link evolution equation this partial differential equation involvesfourspatial dimensions and time the new methods are used to begin investigating the influence of new chemically induced activation link formation and shear induced link breaking in determining when aggregates develop sufficient strength to remain intact and when they are broken apart by fluid stresses

a hybrid high order discretisation of the brinkman problem robust in the darcy and stokes limits abstract in this work we develop and analyse a novel hybrid high order discretisation of the brinkman problem the method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures based on the discrete unknowns we reconstruct inside each element a stokes velocity one degree higher than face unknowns and a darcy velocity in the raviart thomas nedelec space these reconstructed velocities are respectively used to formulate the discrete versions of the stokes and darcy terms in the momentum equation along with suitably designed penalty contributions the proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local stokes or darcy dominated regimes as identified by a dimensionless number which can be interpreted as a friction coefficient the singular limit corresponding to the darcy equation is also fully supported by the method numerical examples corroborate the theoretical results this paper also contains two contributions whose interest goes beyond the specific method and application treated in this work an investigation of the dependence of the constant in the second korn inequality on star shaped domains and its application to the study of the approximation properties of the strain projector in general sobolev seminorms

on matrices having equal corresponding principal minors abstract let a b be n n matrices with entries in a field f we say a and b satisfy property d if b or b t is diagonally similar to a it is clear that if a and b satisfy property d then they have equal corresponding principal minors of all orders the question is to what extent the converse is true there are examples which show the converse is not always true we modify the problem slightly and give conditions on a matrix a which guarantee that if b is any matrix which has the same principal minors as a then a and b will satisfy property d these conditions on a are formulated in terms of ranks of certain submatrices of a and the concept of irreducibility

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

hybridizable discontinuous galerkin method hdg for stokes interface flow abstract in this paper we present a hybridizable discontinuous galerkin hdg method for solving the stokes interface problems with discontinuous viscosity and variable surface tension the jump condition of the stress tensor across the interface is naturally incorporated into the hdg formulation through a constraint on the numerical flux the most important feature of hdg method compared to other dg methods is that it reduces the number of globally coupled unknowns significantly when high order approximate polynomials are used for problems with polygonal interfaces it provides optimal convergence rates of order k 1 in l 2 norm for the velocity pressure and as well as the gradient of velocity furthermore a new approximate velocity can be obtained by an element by element postprocessing which converges with order k 2 in the l 2 norm for stokes interface problems with curved interfaces we use general curvilinear element to ensure the optimal convergence rates an error estimate is given for the approximation of the interface it indicates that curvilinear elements of degree 2 k 1 should be used for optimal convergence rate of order k 1