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Analysis of the Finite Element Method

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___ Introduction

Finite element theory is embedded in a very elegant framework that enables accurate a

priori and a posteriori estimates of discretization errors and convergence rates_ Unfortu_

nately_ a large portion of the theory relies on a knowledge of functional analysis_ which

has not been assumed in this material_ Instead_ we present the relevant concepts and

key results without proof and cite sources of a more complete treatment_ Once again_ we

focus on the model Galerkin problem_ _nd u _ H_

_ satisfying

A_v_ u_ _ _v_ f__ _v _ H_

_ _ _ __a_

where

_v_ f_ _ ZZ

vfdxdy_ _ __b_

A_v_ u_ _ ZZ

_p_vxux _ vyuy_ _ qvudxdy_ _ __c_

where the two_dimensional domain _ has boundary __ _ __E ___N_ For simplicity_ we

have assumed trivial essential and natural boundary data on __E and __N_ respectively_

Finite element solutions U _ SN

_ of _ ___ satisfy

A_V_U_ _ _V_ f__ _V _ SN

_ _ _ ____

where SN

_ is a _nite_dimensional subspace of H_

_ _

As described in Chapter __ error analysis typically proceeds in two steps_

_ Analysis of the Finite Element Method

_ showing that U is optimal in the sense that the error u _ U satis_es

ku _ Uk _ min

W_SN

E ku _Wk _ ____

in an appropriate norm_ and

__ _nding an upper bound for the right_hand side of _ _____

The appropriate norm to use with _ ____ for the model problem _ ___ is the strain

energy norm

kvkA _ pA_v_ v__ _ ____

The _nite element solution might not satisfy _ ____ with other norms and_or problems_

For example_ _nite element solutions are not optimal in any norm for non_self_adjoint

problems_ In these cases_ _ ____ is replaced by the weaker statement

ku _ Uk _ C min

W_SN

_ ku _Wk_ _ ____

C _ _ Thus_ the solution is _nearly best_ in the sense that it only di_ers by a constant

from the best possible solution in the space_

Upper bounds of the right_hand sides of _ ____ or _ ____ are obtained by considering

the error of an interpolant W of u_ Using Theorems _____ and ______ for example_ we

could conclude that

ku _ Wks _ Chp___skukp___ s _ __ _ _ ____

if SN consists of complete piecewise polynomials of degree p with respect to a sequence of

uniform meshes _cf_ De_nition _____ and u _ Hp___ The bound _ ____ can be combined

with either _ ____ or _ ____ to provide an estimate of the error and convergence rate of

a _nite element solution_

The Sobolev norm on H_ and the strain energy norm _ ____ are equivalent for the

model problem _ ___ and we shall use this with _ ____ and _ ____ to construct error

estimates_ Prior to continuing_ you may want to review Sections ____ ____ and ____

A priori _nite element discretization errors_ obtained as described_ do not account for

such _perturbations_ as

_ using numerical integration_

__ interpolating Dirichlet boundary conditions by functions in SN_ and

__ approximating __ by piecewise_polynomial functions_

____ Convergence and Optimality _

These e_ects will have to be appraised_ Additionally_ the a priori error estimates supply

information on convergence rates but are di_cult to use for quantitative error infor_

mation_ A posteriori error estimates_ which use the computed solution_ provide more

practical accuracy appraisals_

___ Convergence and Optimality

While keeping the model problem _ ___ in mind_ we will proceed in a slightly more

general manner by considering a Galerkin problem of the form _ __a_ with a strain

energy A_v_ u_ that is a symmetric bilinear form _cf_ De_nitions ______ __ and is also

continuous and coercive_

De_nition ______ A bilinear form A_v_ u_ is continuous in Hs if there exists a constant

_ _ _ such that

jA_v_ u_j _ _kukskvks_ _u_ v _ Hs_ _ ____

De_nition ______ A bilinear form A_u_ v_ is coercive _Hs _ elliptic or positive de_nite_

in Hs if there exists a constant _ _ _ such that

A_u_ u_ _ _kuk_

_ _u _ Hs_ _ _____

Continuity and coercivity of A_v_ u_ can be used to establish the existence and unique_

ness of solutions to the Galerkin problem _ __a__ These results follow from the Lax_

Milgram Theorem_ We_ll subsequently prove a portion of this result_ but more complete

treatments appear elsewhere ___ __ __ __ We_ll use examples to provide insight into

the meanings of continuity and coercivity_

Example ______ Consider the variational eigenvalue problem_ determine nontrivial

u _ H_

_ and _ _ _____ satisfying

A_u_ v_ _ __u_ v__ _v _ H_

_ _

When A_v_ u_ is the strain energy for the model problem _ ____ smooth solutions of this

variational problem also satisfy the di_erential eigenvalue problem

__pux_x _ _puy_y _ qu _ _u_ _x_ y_ _ __

u _ __ _x_ y_ _ __E_ un _ __ _x_ y_ _ __N_

where n is the unit outward normal to ___

_ Analysis of the Finite Element Method

Letting _r and ur_ r _ _ be an eigenvalue_eigenfunction pair and using the variational

statement with v _ u _ ur_ we obtain the Rayleigh quotient

_r _

A_ur_ ur_

_ur_ ur_

_ r _ _

Since this result holds for all r_ we have

__ _ min

r__

A_ur_ ur_

_ur_ ur_

where __ is the minimum eigenvalue_ _As indicated in Problem _ this result can be

extended__

Using the Rayleigh quotient with _ ______ we have

_r _

_kurk_

s kurk_

_ r _ _

Since kurks _ kurk__ we have

_r _ _ _ __ r _ _

Thus_ _ _ _r_ r _ _ and_ in particular_ _ _ ___

Using _ ____ in conjunction with the Rayleigh quotient implies

_r _

_kurk_

s kurk_

_ r _ _

Combining the two results_

_ kurk_

s kurk_

_ _r _ _kurk_

s kurk_

_ r _ _

Thus_ _ provides a lower bound for the minimum eigenvalue and _ provides a bound for

the maximum growth rate of the eigenvalues in Hs_

Example ______ Solutions of the Dirichlet problem

_uxx _ uyy _ f_x_ y__ _x_ y_ _ __ u _ __ _x_ y_ _ ___

satisfy the Galerkin problem _ ___ with

A_v_ u_ _ ZZ

rv _ rudxdy_ ru _ _ux_ uyT _

An application of Cauchy_s inequality reveals

jA_v_ u_j _ j ZZ

rv _ rudxdyj _ krvk_kruk__

____ Convergence and Optimality _

where

kruk_

_ ZZ

_u_

x _ u_

_dxdy_

Since kruk_ _ kuk__ we have

jA_v_ u_j _ kvk_kuk__

Thus_ _ ____ is satis_ed with s _ and _ _ _ and the strain energy is continuous in

H__

Establishing that A_v_ u_ is coercive in H_ is typically done by using Friedrichs_s _rst

inequality which states that there is a constant _ _ _ such that

kruk_

_ _ _kuk_

_ _ _____

Now_ consider the identity

A_u_ u_ _ kruk_

_ _ __kruk_

and use _ _____ to obtain

A_u_ u_ _ _ __kruk_

_ _ ___kuk_

_ _ _kuk_

where _ _ _ __ max__ ___ Thus_ _ _____ is satis_ed with s _ and A_u_ v_ is coercive

_H__elliptic__

Continuity and coercivity of the strain energy reveal the _nite element solution U to

be nearly the best approximation in SN

Theorem ______ Let A_v_ u_ be symmetric_ continuous_ and coercive_ Let u _ H_

_ satisfy

______a_ and U _ SN

_ H_

_ satisfy ________ Then

ku _ Uk_ _

_ ku _ V k__ _V _ SN

_ _ _ ____a_

with _ and _ satisfying _______ and ________

Remark __ Equation _ ____a_ may also be expressed as

ku _ Uk_ _ C inf

V _SN

_ ku _ V k__ _ ____b_

Thus_ continuity and H__ellipticity give us a bound of the form _ _____

Proof_ cf_ Problem _ at the end of this section_

The bound _ _____ can be improved when A_v_ u_ has the form _ __c__

_ Analysis of the Finite Element Method

Theorem ______ let A_v_ u_ be a symmetric_ continuous_ and coercive bilinear form

u _ H_

_ minimize

I_w _ A_w_w_ _ __w_ f__ _w _ H_

_ _ _ _____

and SN

_ be a _nitedimensional subspace of H_

_ _ Then

__ The minimum of I_W and A_u_W_ u_W__ _W _ SN

_ _ are achieved by the same

function U_

__ The function U is the orthogonal projection of u onto SN

_ with respect to strain

energy_ i_e__

A_V_ u _ U_ _ __ _V _ SN

_ _ _ _____

__ The minimizing function U _ SN

_ satis_es the Galerkin problem

A_V_U_ _ _V_ f__ _V _ SN

_ _ _ ___ _

In particular_ if SN

_ is the whole of H_

A_v_ u_ _ _v_ f__ _v _ H_

_ _ _ _____

Proof_ Our proof will omit several technical details_ which appear in_ e_g__ Wait and

Mitchell ___ Chapter __

Let us begin with _ ___ __ If U minimizes I_W over SN

_ then for any and any

V _ SN

I_U _ I_U _ V _

Using _ ______

I_U _ A_U _ V_ U _ V _ _ __U _ V_ f_

I_U _ I_U _ __A_V_U_ _ _V_ f_ _ _A_V_ V _

_ _ __A_V_U_ _ _V_ f_ _ _A_V_ V __

This inequality must hold for all possible of either sign_ thus_ _ ___ _ must be satis_ed_

Equation _ _____ follows by repeating these arguments with SN

_ replaced by H_

_ _

Next_ replace v in _ _____ by V _ SN

_ H_

_ and subtract _ ___ _ to obtain _ ______

In order to prove Conclusion _ consider the identity

A_u _ U _ V_ u _ U _ V_ _ A_u _ U_ u _ U_ _ _A_u _ U_ V _ _ A_V_ V __

____ Convergence and Optimality

Using _ _____

A_u _ U_ u _ U_ _ A_u _ U _ V_ u _ U _ V _ _ A_V_ V __

Since A_V_ V _ _ __

A_u _ U_ u _ U_ _ A_u _ U _ V_ u _ U _ V __ _V _ SN

_ _

Equality only occurs when V _ __ therefore_ U is the unique minimizing function_

Remark __ We proved a similar result for one_dimensional problems in Theorems

_____ __

Remark __ Continuity and coercivity did not appear in the proof_ however_ they are

needed to establish existence_ uniqueness_ and completeness_ Thus_ we never proved that

limN__ U _ u_ A complete analysis appears in Wait and Mitchell ___ Chapter __

Remark __ The strain energy A_v_ u_ not need be symmetric_ A proof without this

restriction appears in Ciarlet ___

Corollary ______ With the assumptions of Theorem ______

A_u _ U_ u _ U_ _ A_u_ u_ _ A_U_ U__ _ _____

Proof_ cf_ Problem _ at the end of this section_

In Section ____ we obtained a priori estimates of interpolation errors under some

mesh uniformity assumptions_ Recall _cf_ De_nition ______ that we considered a family

of _nite element meshes _h which became _ner as h __ The uniformity condition

implied that all vertex angles were bounded away from _ and _ and that all aspect ratios

were bounded away from _ as h __ Uniformity ensured that transformations from the

physical to the computational space were well behaved_ Thus_ with uniform meshes_ we

were able to show _cf_ Theorem ______ that the error in interpolating a function u _ Hp__

by a complete polynomial W of degree p satis_es

ku _ Wks _ Chp___skukp___ s _ __ _ _ ____a_

The norm on the right can be replaced by the seminorm

juj_

p__ _ X j_j_p__

kD_uk_

_ ____b_

to produce a more precise estimate_ but this will not be necessary for our present appli_

cation_ If singularities are present so that u _ Hq__ with q _ p then_ instead of _ ____a__

we _nd

ku _ Wk_ _ Chqkukq___ _ ____c_

_ Analysis of the Finite Element Method

With optimality _or near optimality_ established and interpolation error estimates

available_ we can establish convergence of the _nite element method_

Theorem ______ Suppose

__ u _ H_

_ and U _ SN

_ H_

_ satisfy _______ and ________ respectively

__ A_v_ u_ is a symmetric_ continuous_ and H_elliptic bilinear form

__ SN

_ consists of complete piecewisepolynomial functions of degree p with respect to

a uniform family of meshes _h and

__ u _ H_

_ _ Hp___

Then

ku _ Uk_ _ Chpkukp__ _ ___a_

and

A_u _ U_ u _ U_ _ Ch_pkuk_

p___ _ ___b_

Proof_ From Theorem ____

A_u _ U_ u _ U_ _ inf

V _SN

A_u _ V_ u _ V _ _ A_u _ W_ u _ W_

where W is an interpolant of u_ Using _ ____ with s _ and v and u replaced by u_W

yields

A_u _ W_ u _W_ _ _ku _Wk_

Using the interpolation estimate _ ____a_ with s _ yields _ ___b__ In order to prove

_ ___a__ use _ _____ with s _ to obtain

_ku _ Uk_

_ _ A_u _ U_ u _ U__

The use of _ ___b_ and a division by _ yields _ ___a__

Since the H_ norm dominates the L_ norm_ _ ___a_ trivially gives us an error esti_

mate in L_ as

ku _ Uk_ _ Chpkukp___

This estimate does not have an optimal rate since the interpolation error _ ____a_ is con_

verging as O_hp____ Getting the correct rate for an L_ error estimate is more complicated

than it is in H__ The proof is divided into two parts_

____ Convergence and Optimality _

Lemma ______ _AubinNitsche_ Under the assumptions of Theorem ______ let __x_ y_ _

_ be the solution of the _dual problem_

A_v_ __ _ _v_ e__ _v _ H_

_ _ _ ____a_

where

e _

u _ U

ku _ Uk_

_ _ ____b_

Let _ _ SN

_ be an interpolant of __ then

ku _ Uk_ _ _ku _ Uk_k_ _ _k__ _ ____c_

Proof_ Set V _ _ in _ _____ to obtain

A___ u _ U_ _ __ _ _____

Take the L_ inner product of _ ____b_ with u _ U to obtain

ku _ Uk_ _ _e_ u _ U__

Setting v _ u _ U in _ ____a_ and using the above relation yields

ku _ Uk_ _ A_u _ U_ ___

Using _ _____

ku _ Uk_ _ A_u _ U_ _ _ ___

Now use the continuity of A_v_ u_ in H_ __ ____ with s _ _ to obtain _ ____c__

Since we have an estimate for ku _ Uk__ estimating ku _ Uk_ by _ ____c_ requires

an estimate of k_ _ _k__ This_ of course_ will be done by interpolation_ however_ use of

_ ____a_ requires knowledge of the smoothness of __ The following lemma provides the

necessary a priori bound_

Lemma ______ Let A_u_ v_ be a symmetric_ H_elliptic bilinear form and u be the solu

tion of _______ on a smooth region __ Then

kuk_ _ Ckfk__ _ _____

Remark __ This result seems plausible since the underlying di_erential equation is of

second order_ so the second derivatives should have the same smoothness as the right_

hand side f_ The estimate might involve boundary data_ however_ we have assumed

trivial conditions_ Let_s further assume that __E is not nil to avoid non_uniqueness

issues_

_ Analysis of the Finite Element Method

Proof_ Strang and Fix ___ Chapter _ establish _ _____ in one dimension_ Johnson ___

Chapter __ obtain a similar result_

With preliminaries complete_ here is the main result_

Theorem ______ Given the assumptions of Theorem ______ then

ku _ Uk_ _ Chp__kukp___ _ _____

Proof_ Applying _ _____ to the dual problem _ ____a_ yields

k_k_ _ Ckek_ _ C_

since kek_ _ according to _ ____b__ With _ _ H__ we may use _ ____c_ with q _ s _

to obtain

k_ _ _k_ _ Chk_k_ _ Ch_

Combining this estimate with _ ___a_ and _ ____c_ yields _ ______

Problems

_ Show that the function u that minimizes

_ _ min

w_H_

_ _ kwk___

A_w_w_

_w_w_

is u__ the eigenfunction corresponding to the minimum eigenvalue __ of A_v_ u_ _

__v_ u__

__ Assume that A_v_ u_ is a symmetric_ continuous_ and H__elliptic bilinear form and_

for simplicity_ that u_ v _ H_

_ _

___ Show that the strain energy and H_ norms are equivalent in the sense that

_kuk_

_ _ A_u_ u_ _ _kuk_

_ _u _ H_

_ _

where _ and _ satisfy _ ____ and _ ______

____ Prove Theorem ____

__ Prove Corollary ___ to Theorem _____

___ Perturbations

In this section_ we examine the e_ects of perturbations due to numerical integration_

interpolated boundary conditions_ and curved boundaries_

____ Perturbations

_____ Quadrature Perturbations

With numerical integration_ we determine U_ as the solution of

A__V_U__ _ _V_ f___ _V _ SN

_ _ _ ___a_

instead of determining U by solving _ ______ The approximate strain energy A__V_U_

or L_ inner product _V_ f__ re ect the numerical integration that has been used_ For

example_ consider the loading

_V_ f_ _

Xe__

_V_ f_e_ _V_ f_e _ ZZ

V _x_ y_f_x_ y_dxdy

where _e is the domain occupied by element e in a mesh of N_ elements_ Using an

n_point quadrature rule _cf_ _____a__ on element e_ we would approximate _V_ f_ by

_V_ f__ _

Xe__

_V_ f_e__ _ ___b_

where

_V_ f_e__ _

Xk__

WkV _xk_ yk_f_xk_ yk__ _ ___c_

The e_ects of transformations to a canonical element have not been shown for simplicity

and a similar formula applies for A__V_U__

Deriving an estimate for the perturbation introduced by _ ___a_ is relatively simple

if A_V_U_ and A__V_U_ are continuous and coercive_

Theorem ______ Suppose that A_v_ u_ and A__V_U_ are bilinear forms with A being

continuous and A_ being coercive in H_ thus_ there exists constants _ and _ such that

jA_u_ v_j _ _kuk_kvk__ _u_ v _ H_

_ _ _ ____a_

and

A__U_ U_ _ _kUk_

_ _U _ SN

_ _ _ ____b_

Then

ku _ U_k_ _ Cfku _ V k_ _ sup

W_SN

jA_V_W_ _ A__V_W_j

kWk_

sup

W_SN

j_W_ f_ _ _W_ f__j

kWk_ g_ _V _ SN

_ _ _ _____

_ Analysis of the Finite Element Method

Proof_ Using the triangular inequality

ku _ U_k_ _ ku _ V _ V _ U_k_ _ ku _ V k_ _ kWk_ _ ____a_

where

W _ U_ _ V_ _ ____b_

Using _ ____b_ and _ ____b_

_kWk_

_ _ A__U_ _ V_W_ _ A__U__W_ _ A__V_W__

Using _ ___a_ with V replaced by W to eliminate A__U__W__ we get

_kWk_

_ _ _f_W__ _ A__V_W__

Adding the exact Galerkin equation _ _____ with v replaced by W

_kWk_

_ _ _f_W__ _ _f_W_ _ A_u_W_ _ A__V_W__

Adding and subtracting A_V_W_ and taking an absolute value

_kWk_

_ _ j_f_W__ _ _f_W_j _ jA_u _ V_W_j _ jA_V_W_ _ A__V_W_j_

Now_ using the continuity condition _ ____a_ with u replaced by u_V and v replaced by

W_ we obtain

_kWk_

_ _ j_f_W__ _ _f_W_j _ _ku _ V k_kWk_ _ jA_V_W_ _ A__V_W_j_

Dividing by _kWk_

kWk_ _

_ f_ku _ V k_ _ j_f_W__ _ _f_W_j

kWk_

_ jA_V_W_ _ A__V_W_j

kWk_ g_

Combining the above inequality with _ ____a__ maximizing the inner product ratios over

W_ and choosing C as the larger of _ _ _ or _ yields _ ______

Remark __ Since the error estimate _ _____ is valid for all V _ SN

_ it can be written

in the form

ku _ U_k_ _ C inf

V _SN

_ fku _ V k_ _ sup

W_SN

jA_V_W_ _ A__V_W_j

kWk_

sup

W_SN

j_W_ f_ _ _W_ f__j

kWk_ g_ _ _____

To bound _ _____ or _ _____ in terms of a mesh parameter h_ we use standard interpola_

tion error estimates _cf_ Sections ___ and ____ for the _rst term and numerical integration

error estimates _cf_ Chapter __ for the latter two terms_ Estimating quadrature errors is

relatively easy and the following typical result includes the e_ects of transforming to a

canonical element_

____ Perturbations _

Theorem ______ Let J__ __ be the Jacobian of a transformation from a computational

__ __plane to a physical _x_ y_plane and let W _ SN

_ _ Relative to a uniform family

of meshes _h_ suppose that det_J__ ___Wx__ __ and det_J__ ___Wy__ __ are piecewise

polynomials of degree at most r_ and det_J__ ___W__ __ is a piecewise polynomial of

degree at most r__ Then

__ If a quadrature rule is exact _in the computational plane_ for all polynomials of

degree at most r_ _ r_

jA_V_W_ _ A__V_W_j

kWk_ _ Chr__kV kr___ _V_W _ SN

_ _ _ ____a_

__ If a quadrature rule is exact for all polynomials of degree at most r_ _ r _ _

j_f_W_ _ _f_W__j

kWk_ _ Chr__kfkr___ _W _ SN

_ _ _ ____b_

Proof_ cf_ Wait and Mitchell ___ Chapter __ or Strang and Fix ___ Chapter __

Example ______ Suppose that the coordinate transformation is linear so that det_J__ ___

is constant and that SN

_ consists of piecewise polynomials of degree at most p_ In this

case_ r_ _ p _ and r_ _ p_ The interpolation error in H_ is

ku _ V k_ _ O_hp__

Suppose that the quadrature rule is exact for polynomials of degree _ or less_ Thus_

_ _ r_ _ r or r _ _ _ p _ and _ ____a_ implies that

jA_V_W_ _ A__V_W_j

kWk_ _ Ch__p__kV k__p___ _V_W _ SN

_ _

With _ _ r_ _ r _ and r_ _ p_ we again _nd r _ _ _ p _ and_ using _ ____b__

j_f_W_ _ _f_W__j

kWk_ _ Ch__p__kfk__p___ _W _ SN

_ _

_ If _ _ __p__ so that r _ p_ then the above perturbation errors are O_hp__ Hence_

all terms in _ _____ or _ _____ have the same order of accuracy and we conclude that

ku _ U_k_ _ O_hp__

This situation is regarded as optimal_ If the coe_cients of the di_erential equation

are constant and_ as is the case here_ the Jacobian is constant_ this result is equiv_

alent to integrating the di_erentiated terms in the strain energy exactly _cf__ e_g__

_ __c___

_ Analysis of the Finite Element Method

_ If _ _ __p _ _ so that r _ p _ then the error in integration is higher order than

the O_hp_ interpolation error_ however_ the interpolation error dominates and

ku _ U_k_ _ O_hp__

The extra e_ort in performing the numerical integration more accurately is not

justi_ed_

_ If _ _ __p _ _ so that r _ p _ then the integration error dominates the interpo_

lation error and determines the order of accuracy as

ku _ U_k_ _ O_h__p____

In particular_ convergence does not occur if _ _ p _ __

Let us conclude this example by examining convergence rates for piecewise_linear _or

bilinear_ approximations _p _ __ In this case_ r_ _ __ r_ _ _ and r _ __ Interpolation

errors converge as O_h__ The optimal order of accuracy of the quadrature rule is _ _ __

i_e__ only constant functions need be integrated exactly_ Performing the integration more

accurately yields no improvement in the convergence rate_

Example ______ Problems with variable Jacobians are more complicated_ Consider

the term

det_J__ ___Wx__ __ _ J_W_x _W__x_

where J _ det_J__ ____ The metrics x and _x are obtained from the inverse Jacobian

J__ _ _ x y

_x _y _ _

J _ y_ _x_

_y_ x_ __

In particular_ x _ y_ J and _x _ _y_ J and

det_J_Wx _ W_y_ _ W_y__

Consider an isoparametric transformation of degree p_ Such triangles or quadrilaterals

in the computational plane have curved sides of piecewise polynomials of degree p in the

physical plane_ If W is a polynomial of degree p then Wx has degree p _ _ Likewise_

x and y are polynomials of degree p in and __ Thus_ y_ and y_ also have degrees

p _ _ Therefore_ JWx and_ similarly_ JWy have degrees r_ _ __p _ __ With J being a

polynomial of degree __p _ __ we _nd JW to be of degree r_ _ _p _ __

For the quadrature errors _ _____ to have the same O_hp_ rate as the interpolation

error_ we must have r _ p_ in _ ____a_b__ Thus_ according to Theorem _____ the order

_ of the quadrature rules in the __ ___plane should be

_ _ r_ _ r _ __p _ _ _ _p _ _ _ __p _ _

____ Perturbations _

for _ ____a_ and

_ _ r_ _ r _ _ __p _ __ _ _p _ _ _ _ __p _ _

for _ ____b__ These results are to be compared with the order of __p _ _ that was

needed with the piecewise polynomials of degree p and linear transformations considered

in Example ____ For quadratic transformations and approximations _p _ ___ we need

third_ and fourth_order quadrature rules for O_h__ accuracy_

_____ Interpolated Boundary Conditions

Assume that integration is exact and the boundary __ is modeled exactly_ but Dirichlet

boundary data is approximated by a piecewise polynomial in SN_ i_e__ by a polynomial

having the same degree p as the trial and test functions_ Under these conditions_ Wait

and Mitchell ___ Chapter __ show that the error in the solution U of a Galerkin problem

with interpolated boundary conditions satis_es

ku _ Uk_ _ Cfhpkukp__ _ hp____kukp__g_ _ ___ _

The _rst term on the right is the standard interpolation error estimate_ The second term

corresponds to the perturbation due to approximating the boundary condition_ As usual_

computation is done on a uniform family of meshes _h and u is smooth enough to be in

Hp___ Brenner and Scott ___ Chapter __ obtain similar results under similar conditions

when interpolation is performed at the Lobatto points on the boundary of an element_

The Lobatto polynomial of degree p is de_ned on ___ as

Lp__ _

dp__

dp__ _ _ __p___ _ ___ _ p _ __

These results are encouraging since the perturbation in the boundary data is of slightly

higher order than the interpolation error_ Unfortunately_ if the domain _ is not smooth

and_ e_g__ contains corners solutions will not be elements of Hp___ Less is known in these

cases_

_____ Perturbed Boundaries

Suppose that the domain _ is replaced by a polygonal domain !_

as shown in Figure

____ Strang and Fix ___ analyze second_order problems with homogeneous Dirichlet

data of the form_ determine u _ H_

_ satisfying

A_v_ u_ _ _v_ f__ _v _ H_

_ _ _ ____a_

_ Analysis of the Finite Element Method

where functions in H_

_ satisfy u_x_ y_ _ __ _x_ y_ _ ___ The _nite element solution

U _ ! SN

_ satis_es

A_V_U_ _ _V_ f__ _V _ ! SN

_ _ _ ____b_

where functions in ! SN

_ vanish on _ !__ _Thus_ ! SN

_ is not a subspace of H_

_ __

Figure ____ Approximation of a curved boundary by a polygon_

For piecewise linear polynomial approximations on triangles they show that ku _

Uk_ _ O_h_ and for piecewise quadratic approximations ku _ Uk_ _ O_h_ ___ The poor

accuracy with quadratic polynomials is due to large errors in a narrow _boundary layer_

near ___ Large errors are con_ned to the boundary layer and results are acceptable

elsewhere_ Wait and Mitchell ___ Chapter __ quote other results which prove that

ku_Uk_ _ O_hp_ for pth degree piecewise polynomial approximations when the distance

between __ and _ !_ is O_hp____ Such is the case when __ is approximated by p th degree

piecewise_polynomial interpolation_

___ A Posteriori Error Estimation

In previous sections of this chapter_ we considered a priori error estimates_ Thus_ we

can_ without computation_ infer that _nite element solutions converge at a certain rate

depending on the exact solution_s smoothness_ Error bounds are expressed in terms of

unknown constants which are di_cult_ if not impossible_ to estimate_ Having computed

a _nite element solution_ it is possible to obtain a posteriori error estimates which give

more quantitative information about the accuracy of the solution_ Many error estimation

techniques are available and before discussing any_ let_s list some properties that a good

a posteriori error estimation procedure should possess_

_ The error estimate should give an accurate measure of the discretization error for

a wide range of mesh spacings and polynomial degrees_

____ A Posteriori Error Estimation

_ The procedure should be inexpensive relative to the cost of obtaining the _nite

element solution_ This usually means that error estimates should be calculated

using only local computations_ which typically require an e_ort comparable to the

cost of generating the sti_ness matrix_

_ A technique that provides estimates of pointwise errors which can subsequently be

used to calculate error measures in several norms is preferable to one that only

works in a speci_c norm_ Pointwise error estimates and error estimates in local

_elemental_ norms may also provide an indications as to where solution accuracy is

insu_cient and where re_nement is needed_

A posteriori error estimates can roughly be divided into four categories_

_ Residual error estimates_ Local _nite element problems are created on either an

element or a subdomain and solved for the error estimate_ The data depends on

the residual of the _nite element solution_

__ Fluxprojection error estimates_ A new ux is calculated by post processing the

_nite element solution_ This ux is smoother than the original _nite element ux

and an error estimate is obtained from the di_erence of the two uxes_

__ Extrapolation error estimates_ Two _nite element solutions having di_erent orders

or di_erent meshes are compared and their di_erences used to provide an error

estimate_

__ Interpolation error estimates_ Interpolation error bounds are used with estimates

of the unknown constants_

The four techniques are not independent but have many similarities_ Surveys of error es_

timation procedures _ _ __ describe many of their properties_ similarities_ and di_erences_

Let us set the stage by brie y describing two simple extrapolation techniques_ Consider a

one_dimensional problem for simplicity and suppose that an approximate solution Up

h_x_

has been computed using a polynomial approximation of degree p on a mesh of spacing

h _Figure _____ Suppose that we have an a priori interpolation error estimate of the

form

u_x_ _ Up

h_x_ _ Cp__hp__ _ O_hp____

We have assumed that the exact solution u_x_ is smooth enough for the error to be

expanded in h to O_hp____ The leading error constant Cp__ generally depends on _un_

known_ derivatives of u_ Now_ compute a second solution with spacing h _ _Figure ____

to obtain

u_x_ _ Up

h___x_ _ Cp___

_p__ _ O_hp____

_ Analysis of the Finite Element Method

Uh/2

Figure ____ Solutions U_

h and U_

h__ computed on meshes having spacing h and h _ with

piecewise linear polynomials _p _ _ and a third solution U_

h computed on a mesh of

spacing h with a piecewise quadratic polynomial _p _ ___

Subtracting the two solutions we eliminate the unknown exact solution and obtain

h___x_ _ Up

h_x_ _ Cp__hp___ _

_p _

_ _ O_hp____

Neglecting the higher_order terms_ we obtain an approximation of the discretization error

Cp__hp__

h___x_ _ Up

h_x_

_ _p__ _

Thus_ we have an estimate of the discretization error of the coarse_mesh solution as

u_x_ _ Up

h_x_

h___x_ _ Up

h_x_

_ _p__ _

The technique is called Richardson_s extrapolation or hextrapolation_ It can also be

used to obtain error estimates of the _ne_mesh solution_ The cost of obtaining the error

estimate is approximately twice the cost of obtaining the solution_ In two and three

dimensions the cost factors rise to_ respectively_ four and eight times the solution cost_

Most would consider this to be excessive_ The only way of justifying the procedure is

to consider the _ne_mesh solution as being the result and the coarse_mesh solution as

furnishing the error estimate_ This strategy only furnishes an error estimate on the coarse

mesh_

Another strategy for obtaining an error estimate by extrapolation is to compute a

second solution using a higher_order method _Figure _____ e_g__

u_x_ _ Up__

h _ Cp__hp__ _ O_hp____

Now_ use the identity

u_x_ _ Up

h_x_ _ _u_x_ _ Up__

h _x_ _ _Up__

h _x_ _ Up

____ A Posteriori Error Estimation _

The _rst term on the right is the O_hp___ error of the higher_order solution and_ hence_

can be neglected relative to the second term_ Thus_ we obtain the approximation

u_x_ _ Up

h_x_ Up__

h _x_ _ Up

h_x__

The di_erence between the lower_ and higher_order solutions furnish an estimate of the er_

ror of the lower_order solution_ The technique is called order embedding or pextrapolation_

There is no error estimate for the higher_order solution_ but some use it without an error

estimate_ This strategy_ called local extrapolation_ can be dangerous near singularities_

Unless there are special properties of the scheme that can be exploited_ the work in_

volved in obtaining the error estimate is comparable to the work of obtaining the solu_

tion_ With a hierarchical embedding_ computations needed for the lower_order method

are also needed for the higher_order method and_ hence_ need not be repeated_

The extrapolation techniques just described are typically too expensive for use as

error estimates_ We_ll develop a residual_based error estimation procedure that follows

Bank _cf_ ___ Chapter _ and uses many of the ideas found in order embedding_ We_ll

follow our usual course of presenting results for the model problem

_r _ pru _ qu _ __pux_x _ _puy_y _ qu _ f_x_ y__ _x_ y_ _ __ _ ___a_

u_x_ y_ _ __ _x_ y_ _ __E_ pun_x_ y_ _ __ _x_ y_ _ __N_ _ ___b_

however_ results apply more generally_ Of course_ the Galerkin form of _ ____ is_ deter_

mine u _ H_

such that

A_v_ u_ _ _v_ f__ _ v__ __ _v _ H_

_ _ _ ____a_

where

_v_ f_ _ ZZ

vfdxdy_ _ ____b_

A_v_ u_ _ ZZ

_prv _ ru _ qvudxdy_ _ ____c_

and

_ v_u __ Z__N

vuds_ _ ____d_

Similarly_ the _nite element solution U _ SN

E H_

satis_es

A_V_U_ _ _V_ f__ _ V__ __ _V _ SN

_ _ _ _____

__ Analysis of the Finite Element Method

We seek an error estimation technique that only requires local _element level_ mesh

computations_ so let_s construct a local Galerkin problem on element e by integrating

_ ___a_ over _e and applying the divergence theorem to obtain_ determine u _ H___e_

such that

Ae_v_ u_ _ _v_ f_e_ _ v_pun _e_ _v _ H___e__ _ ____a_

where

_v_ f_e _ ZZ

vfdxdy_ _ ____b_

Ae_v_ u_ _ ZZ

_prv _ ru _ qvudxdy_ _ ____c_

and

_ v_u _e_ Z__e

vuds_ _ ____d_

As usual_ _e is the domain of element e_ s is a coordinate along __e_ and n is a unit

outward normal to __e_

Let

u _ U _ e_ _ _____

where e_x_ y_ is the discretization error of the _nite element solution_ and substitute

_ _____ into _ ____a_ to obtain

Ae_v_ e_ _ _v_ f_e _ Ae_v_U__ _ v_pun _e_ _v _ H___e__ _ _____

Equation _ ______ of course_ cannot be solved because _i_ v_ u_ and e are elements of an

in_nite_dimensional space and _ii_ the ux pun is unknown on __e_ We could obtain

a _nite element solution of _ _____ by approximating e and v by E and V in a _nite_

dimensional subspace ! SN__e_ of H___e__ Thus_

Ae_V_E_ _ _V_ f_e _ Ae_V_U__ _ V_pun _e_ _V _ ! SN__e__ _ ___ _

We will discuss selection of ! SN momentarily_ Let us _rst prescribe the ux pun

appearing in the last term of _ ___ __ The simplest possibility is to use an average ux

obtained from pUn across the element boundary_ i_e__

Ae_V_E_ _ _V_ f_e _ Ae_V_U__ _ V_

_pUn__ _ _pUn__

_e_ _V _ ! SN__e__ _ _____

____ A Posteriori Error Estimation _

where superscripts _ and __ respectively_ denote values of pUn on the exterior and interior

of __e_

Equation _ _____ is a local Neumann problem for determining the error approximation

E on each element_ No assembly and global solution is involved_ Some investigators prefer

to apply the divergence theorem to the second term on the right to obtain

Ae_V_E_ _ _V_ r_e_ _ V_ _pUn__ _e _ _ V_

_pUn__ _ _pUn__

Ae_V_E_ _ _V_ r_e_ _ V_

_pUn__ _ _pUn__

_e _ ____a_

where

r_x_ y_ _ f _ r _ prU _ qU _ ____b_

is the residual_ This form involves jumps in the ux across element boundaries_

Now let us select the error approximation space ! SN_ Choosing ! SN _ SN does not

work since there are no errors in the solution subspace_ Bank __ chose ! SN as a space of

discontinuous polynomials of the same degree p used for the solution space SN

E _ however_

the algebraic system for E resulting from _ _____ or _ _____ could be ill_conditioned when

the basis is nearly continuous_ A better alternative is to select ! SN as a space of piecewise

p_ st_degree polynomials when SN

E is a space of p th degree polynomials_ Hierarchical

bases _cf_ Sections ___ and ____ are the most e_cient to use in this regard_ Let us

illustrate the procedure by constructing error estimates for a piecewise bilinear solution

on a mesh of quadrilateral elements_ The bilinear shape functions for a canonical _ _ _

square element are

i_j__ __ _ "N

i__ "N

j____ i_ j _ _ __ _ ____a_

where

"N _

_ "N

___ _

_ _ ____b_

The four second_order hierarchical shape functions are

__j__ __ _ "N

j___ "N

_ ___ j _ _ __ _ ___a_

i____ __ _ "N

i__ "N

_ ____ i _ _ __ _ ___b_

where

_ __ _

___ _ _

_p_

_ _ ___c_

__ Analysis of the Finite Element Method

(3,1) (2,1)

(1,3) (2,3)

(1,2) (3,2) (2,2)

(1,1)





Figure _____ Nodal placement for bilinear and hierarchical biquadratic shape functions

on a canonical _ _ _ square element_

Node indexing is given in Figure ____

The restriction of a piecewise bilinear _nite element solution U to the square canonical

element is

U__ __ _

Xi__

Xj__

ijN_

ij__ ___ _ _____

Using either _ _____ or _ ______ the restriction of the error approximation E to the canon_

ical element is the second_order hierarchical function

E__ __ _

Xi__

Xj__

ijN_

ij__ __ _

Xi__

_N_

i___ __ _

Xj__

jN_

_j__ ___ _ _____

The local problems _ _____ or _ _____ are transformed to the canonical element and solved

for the eight unknowns_ c_

ij _ i_ j _ _ __ d_

__ i _ _ __ d_

j_ j _ _ __ using the test functions

V _ Nk

ij _ i_ j _ _ __ __ k _ _ __

Several simpli_cations and variations are possible_ One of these may be called ver

tex superconvergence which implies that the solution at vertices converges more rapidly

than it does globally_ Vertex superconvergence has been rigorously established in certain

circumstances _e_g__ for uniform meshes of square elements__ but it seems to hold more

widely than current theory would suggest_ In the present context_ vertex superconver_

gence implies that the bilinear vertex solution c_

ij _ i_ j _ _ __ converges at a higher rate

than the solution elsewhere on Element e_ Thus_ the error at the vertices c_

ij _ i_ j _ _ __

may be neglected relative to d_

__ i _ _ __ and d_

j_ j _ _ __ With this simpli_cation_

____ A Posteriori Error Estimation __

_ _____ becomes

E__ __ _

Xi__

_N_

i___ __ _

Xj__

jN_

_j__ ___ _ _____

Thus_ there are four unknowns d_

___ d_

___ and d_

__ per element_ This technique may be

carried to higher orders_ Thus_ if SN

E contains complete polynomials of degree p_ ! SN only

contains the hierarchical correction of order p__ All lower_order terms are neglected in

the error estimation space_

The performance of an error estimate is typically appraised in a given norm by com_

puting an e_ectivity index as

# _ kE_x_ y_k

ke_x_ y_k

_ _ _____

Ideally_ the e_ectivity index should not di_er greatly from unity for a wide range of mesh

spacings and polynomial degrees_ Bank and Weiser _ and Oden et al_ _ studied

the error estimation procedure _ _____ with the simplifying assumption _ _____ and were

able to establish upper bounds of the form # _ C in the strain energy norm

kekA__ pA_e_ e__

They could not_ however_ show that the estimation procedure was asymptotically correct

in the sense that # under mesh re_nement or order enrichment_

Example ______ Strouboulis and Haque __ study the properties of several di_erent

error estimation procedures_ We report results for the residual error estimation procedure

_ _____ _____ on the _Gaussian Hill_ problem_ This problem involves a Dirichlet problem

for Poisson_s equation on an equilateral triangle having the exact solution

u_x_ y_ _ __e___ _x___ ___y_______

Errors are shown in Figure ____ for unifom p_re_nement on a mesh of uniform trian_

gular elements having an edge length of ____ and for uniform h_re_nement with p _ __

_Extrapolation_ refers to the p_re_nement procedure described earlier in this section_

This order embedding technique appears to produce accurate error estimates for all poly_

nomial degrees and mesh spacings_ The _residual_ error estimation procedure is _ _____

with errors at vertices neglected and the hierarchical corrections of order p _ forming

! SN _ ______ The procedure does well for even_degree approximations_ but less well for

odd_degree approximations_

From _ ______ we see that the error estimate E is obtained by solving a Neumann

problem_ Such problems are only solvable when the edge loading _the ux average across

__ Analysis of the Finite Element Method

Figure _____ E_ectivity indices for several error estimation procedures using uniform h_

re_nement _left_ and p_re_nement _right_ for the Gaussian Hill Problem __ of Example

____

element edges_ is equilibrated_ The ux averaging used in _ _____ is_ apparently_ not

su_cient to ensure this when p is odd_ We_ll pursue some remedies to this problem later

in this section_ but_ _rst_ let us look at another application_

Example ______ Ai_a __ considers the nonlinear parabolic problem

ut _ qu__u _ _ _

uxx _ uyy

_ _x_ y_ _ ___ _ _ ___ __ t _ __

with the inital and Dirichlet boundary conditions speci_ed so that the exact solution is

u_x_ y_ t_ _

_ epq___x_y_tpq__

He estimates the spatial discretization error using the residual estimate _ _____ neglecting

the error at vertices_ The error estimation space ! SN consists of the hierarchical corrections

of degree p _ _ however_ some lower_degree hierarchical terms are used in some cases_

This is to provide a better equilibration of boundary terms and improve results_ although

this is a time_dependent problem_ which we haven_t studied yet_ Ai_a __ keeps the

temporal errors small to concentrate on spatial error estimation_ With q _ ____ Ai_a_s

__ e_ectivity indices in H_ at t _ ____ are presented in Table ___ for computations

performed on uniform meshes of N_ triangles with polynomial degrees p ranging from

to __

The results with ! SN consisting only of hierarchical corrections of degree p _ are

reasonable_ E_ectivity indices are in excess of ___ for the lower_degree polynomials p _

____ A Posteriori Error Estimation __

p ! SN N_

_ __ __ __

_ ____ ____ ___ ____

_ _ _____ _____ _____ _____

_ _ ____ _____ _____ _____

__ _ __ __ _ __ ___ ____

_ _ _____ __ __ _____ _____

__ _ ____ ___ _____ _____

Table ____ E_ectivity indices in H_ at t _ ____ for Example _____ The degrees of the

hierarchical modes used for ! SN are indicated in that column ___

_ __ but degrade with increasing polynomial degree_ The addition of a lower _third_

degree polynomial correction has improved the error estimates with p _ __ however_

a similar tactic provided little improvement with p _ __ These results and those of

Strouboulis and Haque __ show that the performance of a posteriori error estimates is

still dependent on the problem being solved and on the mesh used to solve it_

Another way of simplifying the error estimation procedure _ _____ and of understand_

ing the di_erences between error estimates for odd_ and even_order _nite element solu_

tions involves a profound_ but little known_ result of Babu$ska _cf_ __ __ __ __ ___ ____

Concentrating on linear second_order elliptic problems on rectangular meshes_ Babu$ska

indicates that asymptotically _as mesh spacing tends to zero_ errors of odd_degree _nite

element solutions occur near element edges while errors of even_degree solutions occur

in element interiors_ These _ndings suggest that error estimates may be obtained by

neglecting errors in element interiors for odd_degree polynomials and neglecting errors

on element boundaries for even_degree polynomials_

Thus_ for piecewise odd_degree approximations_ we could neglect the area integrals

on the right_hand sides of _ _____ or _ ____a_ and calculate an error estimate by solving

Ae_V_E_ __ V_

_pUn__ _ _pUn__

_e_ _V _ ! SN_ _ ____a_

Ae_V_E_ __ V_

_pUn__ _ _pUn__

_e_ _V _ ! SN_ _ ____b_

For piecewise even_degree approximations_ the boundary terms in _ _____ or _ ____a_

can be neglected to yield

Ae_V_E_ _ _V_ f_e _ Ae_V_U__ _V _ ! SN_ _ ___ a_

__ Analysis of the Finite Element Method

Ae_V_E_ _ _V_ r_e_ _V _ ! SN_ _ ___ b_

Yu ____ __ used these arguments to prove asymptotic convergence of error estimates

to true errors for elliptic problems_ Adjerid et al_ ___ _ obtained similar results for

transient parabolic systems_ Proofs_ in both cases_ apply to a square region with square

elements of spacing h _ pN__ A typical result follows_

Theorem ______ Let u _ H_

E _Hp__ and U _ SN

E be solutions of _______ using complete

piecewisebipolynomial functions of order p_

__ If p is an odd positive integer then

ke___ __k_

_ kE___ __k_

_ O_h_p___ _ ____a_

where

kEk_

___p_ _

Xe__

Xi__

Xk__

_Uxi_Pk_e__

_ _ ____b_

Pk_e_ k _ _ __ __ __ are the coordinates of the vertices of _e_ and _f_P_i denotes the

jump in f_x_ in the direction xi_ i _ _ __ at the point P_

__ If p is a positive even integer then _______a_ is satis_ed with

Ae_Vi_ E_ _ _V_ f_e _ Ae_Vi_ U__ _ ____c_

where

E_x__ x__ _ b__e%p__

e _x__ _ b__e%p__

e _x___ _ ____d_

Vi_x__ x__ _ xi

%p__

e _x__

%p__

e _x__

_ i _ _ __ _ ____e_

and %me

_x_ is the mapping of the hierarchical basis function

_ __ _ r_m _

_ Z _

Pm_____d_ _ ____f_

from ___ to the appropriate edge of _e_

Proof_ cf_ Adjerid et al_ ___ _ and Yu ____ ___ Coordinates are written as x _ _x__ x_T

instead of _x_ y_ to simplify notation within summations_ The hierarchical basis element

_ ____f_ is consistent with prior usage_ Thus_ the subscript _ refers to a midside node as

indicated in Figure _____

____ A Posteriori Error Estimation _

Remark __ The error estimate for even_degree approximations has di_erent trial and

test spaces_ The functions Vi_x__ x__ vanish on __e_ Each function is the product of

a _bubble function_ %p__

e _x__%p__

e _x__ biased by a variation in either the x_ or the x_

direction_ As an example_ consider the test functions on the canonical element with

p _ __ Restricting _ ____e_ to the canonical element _ _ __ _ _ _ we have

Vi___ __ _ i

_ ___

_ i _ _ __

Using _ ____f_ with m _ _ or ________

_ __ _

_p_

__ _ __

Thus_

Vi___ __ _

_ _ ___

_ _ __ i _ _ __

Remark __ Theorem ___ applies to tensor_product bi_polynomial bases_ Adjerid et

al_ _ show how this theorem can be modi_ed for use with hierarchical bases_

Example ______ Adjerid et al_ __ solve the nonlinear parabolic problem of Example

____ with q _ __ on uniform square meshes with p ranging from to _ using the error

estimates _ ____a_b_ and _ ____a_c_f__ Temporal errors were controlled to be negligible

relative to spatial errors_ thus_ we need not be concerned that this is a parabolic and not

an elliptic problem_ The exact H_ errors and e_ectivity indices at t _ ___ are presented

in Table _____ Approximate errors are within ten percent of actual for all but one mesh

and appear to be converging at the same rate as the actual errors under mesh re_nement_

p N_ _ __ ___ ___ ___

kek_ kuk_ # kek_ kuk_ # kek_ kuk_ # kek_ kuk_ #

________ _____ _______ ___ _________ _____ _________ _____

_ ___ _____ _____ ________ _____ _________ _____ _________ ____

_ ___ _____ _____ _________ _____ ________ ___ _ _________ ___ _

_ _________ _____ _______ _ ____ ______ _ ____ ________ ____

Table _____ Errors and e_ectivity indices in H_ for Example ____ on N__element uniform

meshes with piecewise bi_p polynomial bases_ Numbers in parentheses indicate a power

of ten_

The error estimation procedures _ _____ and _ _____ use average ux values on __e_

As noted_ data for such _local_ Neumann problems cannot be prescribed arbitrarily_ Let

us examine this further by concentrating on _ _____ which we write as

Ae_V_E_ _ _V_ r_e_ _ V_R _e _ ____a_

__ Analysis of the Finite Element Method

where the elemental residual r was de_ned by _ ____b_ and the boundary residual is

R _ ___pUn__ _ _pUn___ _ ____b_

The function _ on __e was taken as _ to obtain _ ____a__ however_ this may not have

been a good idea for reasons suggested in Example ____

Recall _cf_ Section ___ that smooth solutions of the weak problem _ _____ satisfy

the Neumann problem

_r _ prE _ qE _ r_ _x_ y_ _ _e_ _ _____a_

pEn _ R_ _x_ y_ _ __e_ _ _____b_

Solutions of _ ______ only exist when the data R and r satisfy the equilibrium condition

r_x_ y_dxdy _ Z__e

R_s_ds _ __ _ _____c_

This condition will most likely not be satis_ed by the choice of _ _ __ Ainsworth and

Oden __ describe a relatively simple procedure that requires the solution of the Poisson

problem

___e _ r_ _x_ y_ _ _e_ _ ____a_

__e

_ R_ _x_ y_ _ __e _ __E_ _ ____b_

_e _ __ _x_ y_ _ __E_ _ ____c_

The error estimate is

kEk_

A _

Xe__

Ae__e_ _e__ _ ____d_

The function _ is approximated by a piecewise_linear polynomial in a coordinate s on

__e and may be determined explicitly prior to solving _ ______ Let us illustrate the

e_ect of this equilibrated error estimate_

Example ______ Oden __ considers a _cracked panel_ as shown in Figure ____ and

determines u as the solution of

A_v_ u_ _ ZZ

_vxux _ vyuy_dxdy _ __

____ A Posteriori Error Estimation __

u = 0 y u = 0

u = r1/2 cos 



L

R

Figure _____ Cracked panel used for Example _____

p h #__L_ #__R_ #___

With Without With Without With Without

Balancing Balancing Balancing Balancing Balancing Balancing

__ ___ _____ ___ _ ____ __ ____

__ __ _____ _____ ____ ___ ____

_ __ ___ ___ _ _____ _ _ ____ ____

Table _____ Local and global e_ectivity indices for Example ____ using _ _____ with

and without equilibration_

The essential boundary condition

u_r_ __ _ r___ cos _ _

is prescribed on all boundaries except x _ __ y _ __ Thus_ the solution of the Galerkin

problem will satisfy the natural boundary condition uy _ _ there_ These conditions have

been chosen so that the exact solution is the speci_ed essential boundary condition_ This

solution is singular since ur _ r____ near the origin _r _ ___

Results for the e_ectivity indices in strain energy for the entire region and for the two

elements_ _L and _R_ adjacent to the singularity are shown in Table _____ Computations

were performed on a square grid with uniform spacing h in each coordinate direction

_Figure ______ Piecewise linear and quadratic polynomials were used as _nite element

bases_

Local e_ectivity indices on _L and _R are not close to unity and don_t appear to

be converging as either the mesh spacing is re_ned or p is increased_ Global e_ectivity

indices are near unity_ Convergence to unity is di_cult to appraise with the limited data_

__ Analysis of the Finite Element Method

At this time_ the _eld of a posteriori error estimation is still emerging_ Error estimates

for problems with singularities are not generally available_ The performance of error

estimates is dependent on both the problem_ the mesh_ and the basis_ Error estimates

for realistic nonlinear and transient problems are just emerging_ Verf&urth ___ provides

an exceelent survey of methods and results_

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_ S_ Adjerid_ B_ Belguendouz_ and J_E_ Flaherty_ A posteriori _nite element error

estimation for di_usion problems_ Technical Report ______ Scienti_c Computation

Research Center_ Rensselaer Polytechnic Institute_ Troy_ ____ SIAM Journal on

Scienti_c Computation_ to appear_

__ S_ Adjerid_ J_E_ Flaherty_ and I_ Babu$ska_ A posteriori error estimation for the _nite

element method_of_lines solution of parabolic problems_ Mathematical Models and

Methods in Applied Science_ ____'____ ____

__ S_ Adjerid_ J_E_ Flaherty_ and Y_J_ Wang_ A posteriori error estimation with __

nite element methods of lines for one_dimensional parabolic systems_ Numererishe

Mathematik_ ___'__ ____

__ M_ Ai_a_ Adaptive hpRe_nement Methods for SingularlyPerturbed Elliptic and

Parabolic Systems_ PhD thesis_ Rensselaer Polytechnic Institute_ Troy_ __ _

__ M_ Ainsworth and J_T_ Oden_ A uni_ed approach to a posteriori error estimation

using element residual methods_ Numeriche Mathematik_ _____'___ ____

__ O_ Axelsson and V_A_ Barker_ Finite Element Solution of Boundary Value Problems_

Academic Press_ Orlando_ ____

_ I_ Babu$ska_ T_ Strouboulis_ and C_S_ Upadhyay_ A model study of the quality of

a_posteriori estimators for linear elliptic problems_ Part Ia_ Error estimation in the

interior of patchwise uniform grids of triangles_ Technical Report BN__ _ Institute

for Physical Science and Technology_ University of Maryland_ College Park_ ____

__ I_ Babu$ska_ O_C_ Zienkiewicz_ J_ Gago_ and E_R_ de A_ Oliveira_ editors_ Accuracy

Estimates and Adaptive Re_nements in Finite Element Computations_ John Wiley

and Sons_ Chichester_ ____

__ I_ Babu$ska and D_ Yu_ Asymptotically exact a_posteriori error estimator for bi_

quadratic elements_ Technical Report BN_____ Institute for Physical Science and

Technology_ University of Maryland_ College Park_ ____

__ Analysis of the Finite Element Method

__ R_E_ Bank_ PLTMG A Software Package for Solving Elliptic Partial Di_erential

Equations_ Users_ Guide ____ SIAM_ Philadelphia_ ____

_ R_E_ Bank and A_ Weiser_ Some a posteriori error estimators for elliptic partial

di_erential equations_ Mathematics of Computation_ ______'____ ____

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