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One_Dimensional Finite Element Methods

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___ Introduction

The piecewise_linear Galerkin _nite element method of Chapter _ can be extended in

several directions_ The most important of these is multi_dimensional problems_ however_

we_ll postpone this until the next chapter_ Here_ we_ll address and answer some other

questions that may be inferred from our brief encounter with the method_

__ Is the Galerkin method the best way to construct a variational principal for a partial

di_erential system

_ How do we construct variational principals for more complex problems Speci_cally_

how do we treat boundary conditions other than Dirichlet

__ The _nite element method appeared to converge as O_h in strain energy and O_h_

in L_ for the example of Section ____ Is this true more generally

__ Can the _nite element solution be improved by using higher_degree piecewise_

polynomial approximations What are the costs and bene_ts of doing this

We_ll tackle the Galerkin formulations in the next two sections_ examine higher_degree

piecewise polynomials in Sections __ and ___ and conclude with a discussion of approx_

imation errors in Section ___

___ Galerkin_s Method and Extremal Principles

_For since the fabric of the universe is most perfect and the work of a most

wise creator_ nothing at all takes place in the universe in which some rule of

maximum or minimum does not appear__

One_Dimensional Finite Element Methods

_ Leonhard Euler

Although the construction of variational principles from di_erential equations is an

important aspect of the _nite element method it will not be our main objective_ We_ll

explore some properties of variational principles with a goal of developing a more thorough

understanding of Galerkin_s method and of answering the questions raised in Section ___

In particular_ we_ll focus on boundary conditions_ approximating spaces_ and extremal

properties of Galerkin_s method_ Once again_ we_ll use the model two_point Dirichlet

problem

L_u_ __ __p_xu___ _ q_xu _ f_x_ _ _ x _ __ ____a

u__ _ u__ _ __ ____b

with p_x _ __ q_x _ __ and f_x being smooth functions on _ _ x _ __

As described in Chapter __ the Galerkin form of ____ is obtained by multiplying

____a by a test function v _ H_

_ _ integrating the result on ___ ___ and integrating the

second_order term by parts to obtain

A_v_ u _ _v_ f_ _v _ H_

_ _ ___a

where

_v_ f _ Z _

vfdx_ ___b

and

A_v_ u _ _v__ pu_ _ _v_ qu _ Z _

_v_pu_ _ vqudx_ ___c

and functions v belonging to the Sobolev space H_ have bounded values of

Z _

__v__ _ v__dx_

For _____ a function v is in H_

_ if it also satis_es the trivial boundary conditions

v__ _ v__ _ __ As we shall discover in Section ___ the de_nition of H_

_ will depend on

the type of boundary conditions being applied to the di_erential equation_

There is a connection between self_adjoint di_erential problems such as ____ and

the minimum problem_ _nd w _ H_

_ that minimizes

I_w_ _ A_w_w _ _w_ f _ Z _

_p_w__ _ qw_ _ wf_dx_ ____

____ Galerkin_s Method and Extremal Principles _

Maximum and minimum variational principles occur throughout mathematics and physics

and a discipline called the Calculus of Variations arose in order to study them_ The initial

goal of this _eld was to extend the elementary theory of the calculus of the maxima and

minima of functions to problems of _nding the extrema of functionals such as I_w__ _A

functional is an operator that maps functions onto real numbers_

The construction of the Galerkin form ___ of a problem from the di_erential form

____ is straight forward_ however_ the construction of the extremal problem ____

is not_ We do not pursue this matter here_ Instead_ we refer readers to a text on the

calculus of variations such as Courant and Hilbert ____ Accepting _____ we establish

that the solution u of Galerkin_s method ___ is optimal in the sense of minimizing

_____

Theorem ______ The function u _ H_

_ that minimizes _______ is the one that satis_es

______a_ and conversely_

Proof_ Suppose _rst that u_x is the solution of ___a_ We choose a real parameter _

and any function v_x _ H_

_ and de_ne the comparison function

w_x _ u_x _ _v_x_ ____

For each function v_x we have a one parameter family of comparison functions w_x _ H_

with the solution u_x of ___a obtained when _ _ __ By a suitable choice of _ and

v_x we can use ____ to represent any function in H_

_ _ A comparison function w_x

and its variation _v_x are shown in Figure ____

0 1

v(x)

w(x)

u, w

u(x)

Figure ____ A comparison function w_x and its variation _v_x from u_x_

Substituting ____ into ____

I_w_ _ I_u _ _v_ _ A_u _ _v_ u _ _v _ _u _ _v_ f_

_ One_Dimensional Finite Element Methods

Expanding the strain energy and L_ inner products using ___b_c

I_w_ _ A_u_ u _ _u_ f _ __A_v_ u _ _v_ f_ _ __A_v_ v_

By hypothesis_ u satis_es ___a_ so the O__ term vanishes_ Using _____ we have

I_w_ _ I_u_ _ __A_v_ v_

With p _ _ and q _ __ we have A_v_ v _ __ thus_ u minimizes _____

In order to prove the converse_ assume that u_x minimizes ____ and use ____ to

obtain

I_u_ _ I_u _ _v__

For a particular choice of v_x_ let us regard I_u _ _v_ as a function ____ i_e__

I_u _ _v_ __ ___ _ A_u _ _v_ u _ _v _ _u _ _v_ f_

A necessary condition for a minimum to occur at _ _ _ is ____ _ __ thus_ di_erentiating

____ _ _A_v_ v _ A_v_ u _ _v_ f

and setting _ _ _

____ _ _A_v_ u _ _v_ f_ _ __

Thus_ u is a solution of ___a_

The following corollary veri_es that the minimizing function u is also unique_

Corollary ______ The solution u of ______a_ _or ________ is unique_

Proof_ Suppose there are two functions u__ u_ _ H_

_ satisfying ___a_ i_e__

A_v_ u_ _ _v_ f_ A_v_ u_ _ _v_ f_ _v _ H_

_ _

Subtracting

A_v_ u_ _ u_ _ __ _v _ H_

_ _

Since this relation is valid for all v _ H_

_ _ choose v _ u_ _ u_ to obtain

A_u_ _ u__ u_ _ u_ _ __

If q_x _ __ x _ ___ __ then A_u_ _ u__ u_ _ u_ is positive unless u_ _ u__ Thus_ it

su_ces to consider cases when either _i q_x _ __ x _ ___ ___ or _ii q_x vanishes at

isolated points or subintervals of ___ __ For simplicity_ let us consider the former case_

The analysis of the latter case is similar_

When q_x _ __ x _ ___ ___ A_u_ _ u__ u_ _ u_ can vanish when u_

_ _ u_

_ _ __ Thus_

u_ _ u_ is a constant_ However_ both u_ and u_ satisfy the trivial boundary conditions

____b_ thus_ the constant is zero and u_ _ u__

____ Galerkin_s Method and Extremal Principles _

Corollary ______ If u_w are smooth enough to permit integrating A_u_ v by parts then

the minimizer of ________ the solution of the Galerkin problem ______a__ and the solution

of the two_point boundary value problem _____ _ are all equivalent_

Proof_ Integrate the di_erentiated term in ____ by parts to obtain

I_w_ _ Z _

__w_pw__ _ qw_ _ fw_dx _ wpw_j_

The last term vanishes since w _ H_

__ thus_ using ____a and ___b we have

I_w_ _ _w_L_w_ _ _w_ f_ ____

Now_ follow the steps used in Theorem ___ to show

A_v_ u _ _v_ f _ _v_L_u_ _ f _ __ _v _ H_

_ _

and_ hence_ establish the result_

The minimization problems ____ and ____ are equivalent when w has su_cient

smoothness_ However_ minimizers of ____ may lack the smoothness to satisfy _____

When this occurs_ the solutions with less smoothness are often the ones of physical

interest_

Problems

__ Consider the _stationary value_ problem_ _nd functions w_x that give stationary

values _maxima_ minima_ or saddle points of

I_w_ _ Z _

F_x_w_w_dx ____a

when w satis_es the _essential_ _Dirichlet boundary conditions

w__ _ __ w__ _ __ ____b

Let w _ H_

_ where the subscript E denotes that w satis_es ____b_ and consider

comparison functions of the form ____ where u _ H_

is the function that makes

I_w_ stationary and v _ H_

_ is arbitrary_ _Functions in H_

_ satisfy trivial versions of

____b_ i_e__ v__ _ v__ _ __

Using ____ as an example_ we would have

F_x_w_w_ _ p_x_w__ _ q_xw_ _ wf_x_ _ _ _ _ __

Smooth stationary values of ____ would be minima in this case and correspond

to solutions of the di_erential equation ____a and boundary conditions ____b_

_ One_Dimensional Finite Element Methods

Di_erential equations arising from minimum principles like ____ or from station_

ary value principles like ____ are called Euler_Lagrange equations_

Beginning with _____ follow the steps used in proving Theorem ___ to determine

the Galerkin equations satis_ed by u_ Also determine the Euler_Lagrange equations

for smooth stationary values of _____

___ Essential and Natural Boundary Conditions

The analyses of Section _ readily extend to problems having nontrivial Dirichlet bound_

ary conditions of the form

u__ _ __ u__ _ __ _____a

In this case_ functions u satisfying ___a or w satisfying ____ must be members of

H_ and satisfy _____a_ We_ll indicate this by writing u_w _ H_

_ with the subscript E

denoting that u and w satisfy the essential Dirichlet boundary conditions _____a_ Since

u and w satisfy _____a_ we may use ____ or the interpretation of _v as a variation

shown in Figure ____ to conclude that v should still vanish at x _ _ and _ and_ hence_

belong to H_

_ _

When u is not prescribed at x _ _ and_or __ the function v need not vanish there_

Let us illustrate this when ____a is subject to conditions

u__ _ __ p__u___ _ __ _____b

Thus_ an essential or Dirichlet condition is speci_ed at x _ _ and a Neumann condition is

speci_ed at x _ __ Let us construct a Galerkin form of the problem by again multiplying

____a by a test function v_ integrating on ___ ___ and integrating the second derivative

terms by parts to obtain

Z _

v___pu__ _ qu _ f_dx _ A_v_ u _ _v_ f _ vpu_j_

_ __ ____

With an essential boundary condition at x _ __ we specify u__ _ _ and v__ _ __

however_ u__ and v__ remain unspeci_ed_ We still classify u _ H_

and v _ H_

_ since

they satisfy_ respectively_ the essential and trivial essential boundary conditions speci_ed

with the problem_

With v__ _ _ and p__u___ _ __ we use ____ to establish the Galerkin problem

for ____a_ ____b as_ determine u _ H_

satisfying

A_v_ u _ _v_ f _ v____ _v _ H_

_ _ _____

____ Essential and Natural Boundary Conditions _

Let us reiterate that the subscript E on H_ restricts functions to satisfy Dirichlet _essen_

tial boundary conditions_ but not any Neumann conditions_ The subscript _ restricts

functions to satisfy trivial versions of any Dirichlet conditions but_ once again_ Neumann

conditions are not imposed_

As with problem _____ there is a minimization problem corresponding to _____

determine w _ H_

that minimizes

I_w_ _ A_w_w _ _w_ f _ w____ _____

Furthermore_ in analogy with Theorem ____ we have an equivalence between the Galerkin

_____ and minimization _____ problems_

Theorem ______ The function u _ H_

that minimizes ______ is the one that satis_es

_______ and conversely_

Proof_ The proof is so similar to that of Theorem ___ that we_ll only prove that the

function u that minimizes _____ also satis_es ______ _The remainder of the proof is

stated as Problem _ as the end of this section_

Again_ create the comparison function

w_x _ u_x _ _v_x_ _____

however_ as shown in Figure _____ v__ need not vanish_ By hypothesis we have

u, w

0 1





u(x)

v(x)

w(x)

Figure _____ Comparison function w_x and variation _v_x when Dirichlet data is pre_

scribed at x _ _ and Neumann data is prescribed at x _ __

I_u_ _ I_u _ _v_ _ ___ _ A_u _ _v_ u _ _v _ _u _ _v_ f _ _u__ _ _v_____

_ One_Dimensional Finite Element Methods

Di_erentiating with respect to _ yields the necessary condition for a minimum as

____ _ _A_v_ u _ _v_ f _ v____ _ __

thus_ u satis_es ______

As expected_ Theorem ____ can be extended when the minimizing function u is

smooth_

Corollary ______ Smooth functions u _ H_

satisfying _______ or minimizing ______ also

satisfy _____ a_ ____ b__

Proof_ Using ___c_ integrate the di_erentiated term in _____ by parts to obtain

Z _

v___pu__ _ qu _ f_dx _ v___p__u___ _ __ _ __ _v _ H_

_ _ _____

Since _____ must be satis_ed for all possible test functions_ it must vanish for those

functions satisfying v__ _ __ Thus_ we conclude that ____a is satis_ed_ Similarly_ by

considering test functions v that are nonzero in just a small neighborhood of x _ __ we

conclude that the boundary condition _____b must be satis_ed_ Since _____ must be

satis_ed for all test functions v_ the solution u must satisfy ____a in the interior of the

domain and _____b at x _ __

Neumann boundary conditions_ or other boundary conditions prescribing derivatives

_cf_ Problem at the end of this section_ are called natural boundary conditions be_

cause they follow directly from the variational principle and are not explicitly imposed_

Essential boundary conditions constrain the space of functions that may be used as trial

or comparison functions_ Natural boundary conditions impose no constraints on the

function spaces but_ rather_ alter the variational principle_

Problems

__ Prove the remainder of Theorem _____ i_e__ show that functions that satisfy _____

also minimize ______

_ Show that the Galerkin form ____a with the Robin boundary conditions

p__u___ _ __u__ _ ___ p__u___ _ __u__ _ __

is_ determine u _ H_ satisfying

A_v_ u _ _v_ f _ v_____ _ __u__ _ v_____ _ __u___ _v _ H__

Also show that the function w _ H_ that minimizes

I_w_ _ A_w_w _ _w_ f _ __w__ _ __w___ _ __w__ _ __w___

is u_ the solution of the Galerkin problem_

____ Piecewise Lagrange Polynomials _

__ Construct the Galerkin form of ____ when

p_x _ _ __ if _ _ x _ _

_ if _ _ x _ _

Such a situation can arise in a steady heat_conduction problem when the medium

is made of two di_erent materials that are joined at x _ _ _ What conditions

must u satisfy at x _ _

___ Piecewise Lagrange Polynomials

The _nite element method is not limited to piecewise_linear polynomial approximations

and its extention to higher_degree polynomials is straight forward_ There is_ however_ a

question of the best basis_ Many possibilities are available from design and approximation

theory_ Of these_ splines and Hermite approximations ___ are generally not used because

they o_er more smoothness and_or a larger support than needed or desired_ Lagrange

interpolation __ and a hierarchical approximation in the spirit of Newton_s divided_

di_erence polynomials will be our choices_ The piecewise_linear _hat_ function

j_x ___

x_xj__

xj_xj__

_ if xj__ _ x _ xj

xj___x

xj___xj

_ if xj _ x _ xj__

__ otherwise

_____a

on the mesh

x_ _ x_ _ _ _ _ _ xN _____b

is a member of both classes_ It has two desirable properties_ _i j_x is unity at node

j and vanishes at all other nodes and _ii j is only nonzero on those elements contain_

ing node j_ The _rst property simpli_es the determination of solutions at nodes while

the second simpli_es the solution of the algebraic system that results from the _nite

element discretization_ The Lagrangian basis maintains these properties with increasing

polynomial degree_ Hierarchical approximations_ on the other hand_ maintain only the

second property_ They are constructed by adding high_degree corrections to lower_degree

members of the series_

We will examine Lagrange bases in this section_ beginning with the quadratic poly_

nomial basis_ These are constructed by adding an extra node xj____ at the midpoint of

each element _xj___ xj __ j _ __ _ _ _ _ _N _Figure _____ As with the piecewise_linear basis

_____a_ one basis function is associated with each node_ Those associated with vertices

are

__ One_Dimensional Finite Element Methods

x x 0 x x x x 2 xN-1

U(x)

1/2 1 3/2 N-1/2 N x

Figure _____ Finite element mesh for piecewise_quadratic Lagrange polynomial approxi_

mations_

j_x _ __

_ _ __x_xj

_ _x_xj

__ if xj__ _ x _ xj

_ _ __x_xj

hj__

_ _x_xj

hj__

__ if xj _ x _ xj__

__ otherwise

_ j _ __ __ _ _ _ _N_ ____a

and those associated with element midpoints are

j_____x _ _ _ _ __x_xj____

__ if xj__ _ x _ xj

__ otherwise

_ j _ __ _ _ _ _ _N_ ____b

Here

hj _ xj _ xj___ j _ __ _ _ _ _ _N_ ____c

These functions are shown in Figure ____ Their construction _to be described invovles

satsifying

j_xk _ _ __ if j _ k

__ otherwise

_ j_ k _ __ _ _ __ _ _ _ _N _ __N _ _ _N_ _____

Basis functions associated with a vertex are nonzero on at most two elements and those

associated with an element midpoint are nonzero on only one element_ Thus_ as noted_

the Lagrange basis function j is nonzero only on elements containing node j_ The

functions ____a_b are quadratic polynomials on each element_ Their construction and

trivial extension to other _nite elements guarantees that they are continuous over the

entire mesh and_ like ______ are members of H__

The _nite element trial function U_x is a linear combination of ____a_b over the

vertices and element midpoints of the mesh that may be written as

U_x _

Xj__

cjj_x _

Xj__

cj____j_____x _

Xj__

cj__j___x_ _____

____ Piecewise Lagrange Polynomials __

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.2

0.2

0.4

0.6

0.8

1.2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure ____ Piecewise_quadratic Lagrange basis functions for a vertex at x _ _ _left and

an element midpoint at x _ ____ _right_ When comparing with _____ set xj__ _ ___

xj____ _ _____ xj _ __ xj____ _ ____ and xj__ _ __

Using ______ we see that U_xk _ ck_ k _ __ _ _ __ _ _ _ _N _ _ _N_

Cubic_ quartic_ etc_ Lagrangian polynomials are generated by adding nodes to element

interiors_ However_ prior to constructing them_ let_s introduce some terminology and

simplify the node numbering to better suit our task_ Finite element bases are constructed

implicitly in an element_by_element manner in terms of shape functions_ A shape function

is the restriction of a basis function to an element_ Thus_ for the piecewise_quadratic

Lagrange polynomial_ there are three nontrivial shape functions on the element _j __

_xj___ xj __

_ the right portion of j___x

Nj___j_x _ _ _ __

x _ xj__

_ _

x _ xj__

__ _____a

_ j_____x

Nj_____j_x _ _ _ __

x _ xj____

__ _____b

_ and the left portion of j_x

Nj_j_x _ ____

x _ xj

_ _

x _ xj

__ x _ _j _ _____c

_Figure _____ In these equations_ Nk_j is the shape function associated with node k_

k _ j _ __ j _ _ _ j_ of element j _the subinterval _j_ We may use _____ and _____

to write the restriction of U_x to _j as

U_x _ cj__Nj___j _ cj____Nj_____j _ cjNj_j_ x _ _j _

_ One_Dimensional Finite Element Methods

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2

0.2

0.4

0.6

0.8

1.2

Figure _____ The three quadratic Lagrangian shape functions on the element _xj___ xj __

When comparing with ______ set xj__ _ __ xj____ _ ____ and xj _ __

More generally_ we will associate the shape function Nk_e_x with mesh entity k of

element e_ At present_ the only mesh entities that we know of are vertices and _nodes

on elements_ however_ edges and faces will be introduced in two and three dimensions_

The key construction concept is that the shape function Nk_e_x is

__ nonzero only on element e and

_ nonzero only if mesh entity k belongs to element e_

A one_dimensional Lagrange polynomial shape function of degree p is constructed

on an element e using two vertex nodes and p _ _ nodes interior to the element_ The

generation of shape functions is straight forward_ but it is customary and convenient to

do this on a _canonical element__ Thus_ we map an arbitrary element _e _ _xj___ xj _

onto __ _ _ _ _ by the linear transformation

x__ _

_ _ _

xj__ _

_ _ _

xj _ _ _ ____ ___ _____

Nodes on the canonical element are numbered according to some simple scheme_ i_e__ _

to p with __ _ ___ _p _ __ and _ _ __ _ __ _ _ _ _ _ _p__ _ _ _Figure _____ These are

mapped to the actual physical nodes xj___ xj_____p_ _ _ _ _ xj on _e using ______ Thus_

xj___i_p _

_ _ _i

xj__ _

_ _ _i

xj _ i _ __ __ _ _ _ _ p_

____ Piecewise Lagrange Polynomials __





1

−0 k N

k,e

Figure _____ An element e used to construct a p th_degree Lagrangian shape function

and the shape function Nk_e_x associated with node k_

The Lagrangian shape function Nk_e__ of degree p has a unit value at node k of

element e and vanishes at all other nodes_ thus_

Nk_e__l _ _kl _ _ __ if k _ l

__ otherwise

_ l _ __ __ _ _ _ _ p_ _____a

It is extended trivially when _ _ ____ ___ The conditions expressed by _____a imply that

Nk_e__ _

Y l___ l__k

_ _ _l

_k _ _l

__ _ ____ _ __ _ _ _ __ _ _k____ _ _k__ _ _ _ __ _ _p

__k _ ____k _ __ _ _ _ __k _ _k____k _ _k__ _ _ _ __k _ _p

_____b

We easily check that Nk_e _i is a polynomial of degree p in _ and _ii it satis_es conditions

_____a_ It is shown in Figure _____ Written in terms of shape function_ the restriction

of U to the canonical element is

U__ _

Xk__

ckNk_e___ _____

Example ___ _ Let us construct the quadratic Lagrange shape functions on the

canonical element by setting p _ in _____b to obtain

N__e__ _

__ _ ____ _ __

___ _ _____ _ __

_ N__e__ _

__ _ ____ _ __

___ _ _____ _ __

N__e__ _

__ _ ____ _ __

___ _ _____ _ __

__ One_Dimensional Finite Element Methods

Setting __ _ ___ __ _ __ and __ _ _ yields

N__e__ _

___ _ _

_ N__e__ _ __ _ ___ N__e__ _

__ _ __

_ _____

These may easily be shown to be identical to ____ by using the transformation _____

_see Problem _ at the end of this section_

Example _____ Setting p _ _ in _____b_ we obtain the linear shape functions on the

canonical element as

N__e _

_ _ _

_ N__e _

_ _ _

_ ______

The two nodes needed for these shape functions are at the vertices __ _ __ and __ _ __

Using the transformation ______ these yield the two pieces of the hat function _____a_

We also note that these shape functions were used in the linear coordinate transformation

______ This will arise again in Chapter __

Problems

__ Show the the quadratic Lagrange shape functions _____ on the canonical ____ __

element transform to those on the physical element ____ upon use of _____

_ Construct the shape functions for a cubic Lagrange polynomial from the general

formula _____ by using two vertex nodes and two interior nodes equally spaced on

the canonical ____ __ element_ Sketch the shape functions_ Write the basis functions

for a vertex and an interior node_

___ Hierarchical Bases

With a hierarchical polynomial representation the basis of degree p _ _ is obtained as a

correction to that of degree p_ Thus_ the entire basis need not be reconstructed when

increasing the polynomial degree_ With _nite element methods_ they produce algebraic

systems that are less susceptible to round_o_ error accumulation at high order than those

produced by a Lagrange basis_

With the linear hierarchical basis being the usual hat functions ______ let us begin

with the piecewise_quadratic hierarchical polynomial_ The restriction of this function to

element _e _ _xj___ xj _ has the form

U__x _ U__x _ cj____N_

j_____e_x_ x _ _e_ _____a

where U__x is the piecewise_linear _nite element approximation on _e

U__x _ cj__N_

j___e_x _ cjN_

j_e_x_ _____b

____ Hierarchical Bases __

Superscripts have been added to U and Nj_e to identify their polynomial degree_ Thus_

j___e_x _ _ xj_x

_ if x _ _e

__ otherwise

_ _____c

j_e_x _ _ x_xj__

_ if x _ _e

__ otherwise

_____d

are the usual hat function _____ associated with a piecewise_linear approximation U__x_

The quadratic correction N_

j_____e_x is required to _i be a quadratic polynomial_ _ii

vanish when x _ _e_ and _iii be continuous_ These conditions imply that N_

j_____e is

proportional to the quadratic Lagrange shape function _____b and we will take it to be

identical_ thus_

j_____e_x _ _ _ _ __x_xj____

__ if x _ _e

__ otherwise

_ _____e

The normalization N_

j_____e_xj____ _ _ is not necessary_ but seems convenient_

Like the quadratic Lagrange approximation_ the quadratic hierarchical polynomial has

three nontrivial shape functions per element_ however_ two of them are linear and only

one is quadratic _Figure _____ The basis_ however_ still spans quadratic polynomials_

Examining ______ we see that cj__ _ U_xj__ and cj _ U_xj_ however_

U_xj____ _

cj__ _ cj

_ cj_____

Di_erentiating _____a twice with respect to x gives an interpretation to cj____ as

cj____ _ _

U___xj_____

This interpretation may be useful but is not necessary_

A basis may be constructed from the shape functions in the manner described for

Lagrange polynomials_ With a mesh having the structure used for the piecewise_quadratic

Lagrange polynomials _Figure _____ the piecewise_quadratic hierarchical functions have

the form

U_x _

Xj__

cj_

_x _

Xj__

cj_____

_____x ____

where _

_x is the hat function basis _____a and _

_x _ N_

j_e_x_

Higher_degree hierarchical polynomials are obtained by adding more correction terms

to the lower_degree polynomials_ It is convenient to construct and display these poly_

nomials on the canonical ____ __ element used in Section ___ The linear transformation

__ One_Dimensional Finite Element Methods

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure _____ Quadratic hierarchical shape on _xj___ xj __ When comparing with ______

set xj__ _ _ and xj _ __

_____ is again used to map an arbitrary element _xj___ xj_ onto __ _ _ _ __ The vertex

nodes at _ _ __ and _ are associated with the linear shape functions and_ for simplicity_

we will index them as __ and __ The remaining p__ shape functions are on the element

interior_ They need not be associated with any nodes but_ for convenience_ we will asso_

ciate all of them with a single node indexed by _ at the center __ _ _ of the element_

The restriction of the _nite element solution U__ to the canonical element has the form

U__ _ c__N_

____ _ c_N_

_ __ _

Xi__

ciNi

___ _ _ ____ ___ _____

_We have dropped the elemental index e on Ni

j_e since we are only concerned with ap_

proximations on the canonical element_ The vertex shape functions N_

_ and N_

_ are the

hat function segments ______ on the canonical element

____ _

_ _ _

_ N_

_ __ _

_ _ _

_ _ _ ____ ___ _____

Once again_ the higher_degree shape functions Ni_

___ i _ _ __ _ _ _ _ p_ are required to have

the proper degree and vanish at the element_s ends _ _ ___ _ to maintain continuity_

Any normalization is arbitrary and may be chosen to satisfy a speci_ed condition_ e_g__

___ _ __ We use a normalization of Szab o and Babu!ska ___ which relies on Legendre

polynomials_ The Legendre polynomial Pi___ i _ __ is a polynomial of degree i in _

satisfying ____

____ Hierarchical Bases __

__ the di_erential equation

__ _ __P__

i _ _P_

i _ i_i _ _Pi _ __ __ _ _ _ __ i _ __ _____a

_ the normalization

Pi__ _ __ i _ __ _____b

__ the orthogonality relation

Z _

Pi__Pj__d_ _

i _ _ _ __ if i _ j

__ otherwise

_ _____c

__ the symmetry condition

Pi___ _ ___iPi___ i _ __ _____d

__ the recurrence relation

_i _ _Pi____ _ _i _ __Pi__ _ iPi_____ i _ __ _____e

and

__ the di_erentiation formula

i____ _ _i _ _Pi__ _ P_

i_____ i _ __ _____f

The _rst six Legendre polynomials are

P___ _ __ P___ _ __

P___ _

___ _ _

_ P___ _

___ _ __

P___ _

____ _ ____ _ _

_ P___ _

____ _ ____ _ ___

_ _____

With these preliminaries_ we de_ne the shape functions

Ni_

__ _ ri _ _

Z _

Pi___d_ i _ _ _____a

Using _____d_f_ we readily show that

Ni_

__ _

Pi__ _ Pi____

p_i _ _

_ i _ _ _____b

__ One_Dimensional Finite Element Methods

Use of the normalization and symmetry properties _____b_d further reveal that

Ni_

___ _ Ni_

__ _ __ i _ _ _____c

and use of the orthogonality property _____c indicates that

Z _

dNi_

dNj

_ __

d_ _ _ij _ i_ j _ _ _____d

Substituting _____ into _____b gives

_ __ _

___ _ __ N_

_ __ _

p__

____ _ __

_ __ _

_p__

____ _ ___ _ __ N_

_ __ _

_p__

____ _ ____ _ ___ _____

Shape functions Ni_

___ i _ _ __ _ _ _ _ __ are shown in Figure ____

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

Figure ____ One_dimensional hierarchical shape functions of degrees _solid_ ____ _

_ _ _ ___ and _ _" on the canonical element __ _ _ _ __

The representation _____ with use of _____b_d reveals that the parameters c__ and

c_ correspond to the values of U___ and U___ respectively_ however_ the remaining

parameters ci_ i _ _ do not correspond to solution values_ In particular_ using ______

____ Hierarchical Bases __

_____d_ and _____b yields

U__ _

c__ _ c_

Xi____

ciNi_

___

Hierarchical bases can be constructed so that ci is proportional to diU__ d_i_ i _

_cf_ ____ Section ___ however_ the shape functions _____ based on Legendre polynomials

reduce sensitivity of the basis to round_o_ error accumulation_ This is very important

when using high_order _nite element approximations_

Example ____ _ Let us solve the two_point boundary value problem

_pu__ _ qu _ f_x_ _ _ x _ __ u__ _ u__ _ __ _____

using the _nite element method with piecewise_quadratic hierarchical approximations_

As in Chapter __ we simplify matters by assuming that p _ _ and q _ _ are constants_

By now we are aware that the Galerkin form of this problem is given by ____ As

in Chapter __ introduce _cf_ ______

ASj

_v_ u _ Z xj

xj__

pv_u_dx_

We use _____ to map _xj___ xj_ to the canonical ____ __ element as

ASj

_v_ u _

hj Z _

d__ ______

Using ______ we write the restriction of the piecewise_quadratic trial and text functions

to _xj___ xj _ as

U__ _ _cj___ cj_ cj______

_ V __ _ _dj___ dj_ dj______

_ ______

Substituting ______ into ______

ASj

_V_U _ _dj___ dj_ dj_____Kj_

cj__

cj____

_____a

where Kj is the element sti_ness matrix

Kj _

hj Z _

d__

_N_

___N_

__N_

_ _d__

_ One_Dimensional Finite Element Methods

Substituting for the basis de_nitions ______ ____

Kj _

hj Z _

___

_q_

___ _ _ _ _r_

_d__

Integrating

Kj _

hj Z _

___

_ _ __ _ __p_ _

__ _ _ _ _p_ _

__p_ _ _p_ _ ___

d_ _

hj_

_ __ _

__ _ _

_ _

_ _____b

The orthogonality relation _____d has simpli_ed the sti_ness matrix by uncoupling the

linear and quadratic modes_

In a similar manner_

AMj

_V_U _ Z xj

xj__

qV Udx _

qhj

Z _

V Ud__ ______a

Using ______

AMj

_V_U _ _dj___ dj_ dj_____Mj_

cj__

cj____

where_ upon use of ______ _____ the element mass matrix Mj satis_es

Mj _

qhj

Z _

___

_N_

___N_

__N_

_ _d_ _

qhj

_ _p_

_p_ _p_ _ _

______c

The higher and lower order terms of the element mass matrix have not decoupled_ Com_

paring _____b and ______c with the forms developed in Section ___ for piecewise_linear

approximations_ we see that the piecewise linear sti_ness and mass matrices are contained

as the upper portions of these matrices_ This will be the case for linear problems_

thus_ each higher_degree polynomial will add a _border_ to the lower_degree sti_ness and

mass matrices_

Finally_ consider

_V_ fj _ Z xj

xj__

V fdx _

Z _

V fd__ ______a

Using ______

_V_ fj _ _dj___ dj_ dj_____lj ______b

____ Hierarchical Bases _

where

lj _

Z _

___

f_x__d__ ______c

As in Section ____ we approximate f_x by piecewise_linear interpolation_ which we write

f_x N_

____fj__ _ N_

_ __fj

with fj __ f_xj_ The manner of approximating f_x should clearly be related to the

degree p and we will need a more careful analysis_ Postponing this until Chapters _ and

__ we have

lj _

Z _

___

_N_

___N_

_ _d_ fj__

fj _ _

fj__ _ fj

_p_ _fj__ _ fj

______d

Using ___a with _____a_ ______a_ and ______a_ we see that assembly requires

evluating the sum

Xj__

_ASj

_V_U _ AMj

_V_U _ _V_ fj_ _ __

Following the strategy used for the piecewise_linear solution of Section ____ the local

sti_ness and mass matrices and load vectors are added into their proper locations in

their global counterparts_ Imposing the condition that the system be satis_ed for all

choices of dj_ j _ _ _ __ _ _ _ _ _ _N _ __ yields the linear algebraic system

_K _Mc _ l_ ______

The structure of the sti_ness and mass matrices K and M and load vector l depend on

the ordering of the unknowns c and virtual coordinates d_ One possibility is to order

them by increasing index_ i_e__

c _ _c____ c__ c____ c__ _ _ _ _ cN___ cN_____T _ ______

As with the piecewise_linear basis_ we have assumed that the homogeneous boundary

conditions have explicitly eliminated c_ _ cN _ __ Assembly for this ordering is similar

to the one used in Section ___ _cf_ Problem at the end of this section_ This is a natural

ordering and the one most used for this approximation_ however_ for variety_ let us order

the unknowns by listing the vertices _rst followed by those at element midpoints_ i_e__

c _ cL

cQ __ cL __

____

___

cN__

___

_ cQ __

____

c___

___

cN____

___

_ ______

One_Dimensional Finite Element Methods

In this case_ K_ M_ and l have a block structure and may be partitioned as

K _ KL _

_ KQ __ M _ ML MLQ

LQ MQ __ l _ lL

lQ _ ______

where_ for uniform mesh spacing hj _ h_ j _ __ _ _ _ _ _N_ these matrices are

KL _

_______

__ __

_ _ _

__ __

_____

_ KQ _

_______

_ _ _

_____

_ ______

ML _

_______

_ _

_ _ _

_ _

_____

_ MLQ _ _

_ r_

_______

_ _

_ _ _

_ _

_____

MQ _

_______

_ _ _

_____

_ _____

lL _

_____

f_ _ _f_ _ f_

___

fN__ _ _fN__ _ fN

___

_ lQ _ _

_____

f_ _ f_

___

fN__ _ fN

___

_ _____

With N _ _ vertex unknowns cL and N elemental unknowns cQ_ the matrices KL and

ML are _N __ _N ___ KQ and MQ are N N_ and MLQ is _N __ N_ Similarly_

lL and lQ have dimension N __ and N_ respectively_ The indicated ordering implies that

the _ _ element sti_ness and mass matrices _____b and ______c for element j are

added to rows and columns j _ __ j_ and N _ _ _ j of their global counterparts_ The

_rst row and column of the element sti_ness and mass matrices are deleted when j _ _

to satisfy the left boundary condition_ Likewise_ the second row and column of these

matrices are deleted when j _ N to satisfy the right boundary condition_

The structure of the system matrix K _M is

K _M _ KL _ML MLQ

LQ KQ _MQ __ ____

____ Hierarchical Bases _

The matrix KL _ ML is the same one used for the piecewise_linear solution of this

problem in Section ____ Thus_ an assembly and factorization of this matrix done during a

prior piecewise_linear _nite element analysis could be reused_ A solution procedure using

this factorization is presented as Problem _ at the end of this section_ Furthermore_ if

q _ _ then MLQ _ _ _cf_ _____b and the linear and quadratic portions of the system

uncouple_

In Example ______ we solved _____ with p _ __ q _ __ and f_x _ x using piecewise_

linear _nite elements_ Let us solve this problem again using piecewise_quadratic hier_

archical approximations and compare the results_ Recall that the exact solution of this

problem is

u_x _ x _

sinh x

sinh _

Results for the error in the L_ norm are shown in Table ____ for solutions obtained

with piecewise_linear and quadratic approximations_ The results indicate that solutions

with piecewise_quadratic approximations are converging as O_h_ as opposed to O_h_

for piecewise_linear approximations_ Subsequently_ we shall show that smooth solutions

generally converge as O_hp__ in the L_ norm and as O_hp in the strain energy _or H_

norm_

N Linear Quadratic

DOF jjejj_ jjejj_ h_ DOF jjejj_ jjejj_ h_

_ _ ______ _______ _ _______ _______

_ _ ________ _______ __ ________ _______

__ __ ________ _______ __ ________ _______

_ __ ________ _______

Table _____ Errors in L_ and degrees of freedom _DOF for piecewise_linear and piecewse_

quadratic solutions of Example _____

The number of elements N is not the only measure of computational complexity_

With higher_order methods_ the number of unknowns _degrees of freedom provides a

better index_ Since the piecewise_quadratic solution has approximately twice the number

of unknowns of the linear solution_ we should compare the linear solution with spacing h

and the quadratic solution with spacing h_ Even with this analysis_ the superiority of

the higher_order method in Table ____ is clear_

Problems

__ Consider the approximation in strain energy of a given function u___ __ _ _ _ __

by a polynomial U__ in the hierarchical form ______ The problem consists of

_ One_Dimensional Finite Element Methods

determining U__ as the solution of the Galerkin problem

A_V_U _ A_V_ u_ _V _ Sp_

where Sp is a space of p th_degree polynomials on ____ ___ For simplicity_ let us take

the strain energy as

A_v_ u _ Z _

v_u_d__

With c__ _ u___ and c_ _ u___ _nd expressions for determining the remaining

coe_cients ci_ i _ _ __ _ _ _ _ p_ so that the approximation satis_es the speci_ed

Galerkin projection_

_ Show how to generate the global sti_ness and mass matrices and load vector for

Example ____ when the equations and unknowns are written in order of increasing

index _______

__ Suppose KL _ML have been assembled and factored by Gaussian elimination as

part of a _nite element analysis with piecewise_linear approximations_ Devise an

algorithm to solve ______ for cL and cQ that utilizes the given factorization_

___ Interpolation Errors

Errors of _nite element solutions can be measured in several norms_ We have already

introduced pointwise and global metrics_ In this introductory section on error analysis_

we_ll de_ne some basic principles and study interpolation errors_ As we shall see shortly_

errors in interpolating a function u by a piecewise polynomial approximation U will

provide bounds on the errors of _nite element solutions_

Once again_ consider a Galerkin problem for a second_order di_erential equation_ _nd

u _ H_

_ such that

A_v_ u _ _v_ f_ _v _ H_

_ _ _____

Also consider its _nite element counterpart_ _nd U _ SN

_ such that

A_V_U _ _V_ f_ _V _ SN

_ _ ____

Let the approximating space SN

_ _ H_

_ consist of piecewise_polynomials of degree p on

N_element meshes_ We begin with two fundamental results regarding Galerkin_s method

and _nite element approximations_

____ Interpolation Errors _

Theorem ______ Let u _ H_

_ and U _ SN

_ _ H_

_ satisfy _____ _ and ________ respectively_

then

A_V_ u _ U _ __ _V _ SN

_ _ _____

Proof_ Since V _ SN

_ it also belongs to H_

_ _ Thus_ it may be used to replace v in ______

Doing this and subtracting ____ yields the result_

We shall subsequently show that the strain energy furnishes an inner product_ With

this interpretation_ we may regard _____ as an orthogonality condition in a _strain

energy space_ where A_v_ u is an inner product and pA_u_ u is a norm_ Thus_ the

_nite element solution error

e_x __ u_x _ U_x _____

is orthogonal in strain energy to all functions V in the subspace SN

_ _ We use this orthog_

onality to show that solutions obtained by Galerkin_s method are optimal in the sence of

minimizing the error in strain energy_

Theorem ______ Under the conditions of Theorem ____ _

A_u _ U_ u _ U _ min

V _SN

A_u _ V_ u _ V _ _____

Proof_ Consider

A_u _ U_ u _ U _ A_u_ u _ A_u_ U _ A_U_ U_

Use _____ with V replaced by U to write this as

A_u _ U_ u _ U _ A_u_ u _ A_u_ U _ A_U_ U _ A_u _ U_ U

A_u _ U_ u _ U _ A_u_ u _ A_U_ U_

Again_ using _____ for any V _ SN

A_u _ U_ u _ U _ A_u_ u _ A_U_ U _ A_V_ V _ A_V_ V _ A_u _ U_ V

A_u _ U_ u _ U _ A_u _ V_ u _ V _ A_U _ V_U _ V _

Since the last term on the right is non_negative_ we can drop it to obtain

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

_ _

We see that equality is attained when V _ U and_ thus_ _____ is established_

_ One_Dimensional Finite Element Methods

With optimality of Galerkin_s method_ we may obtain estimates of _nite element

discretization errors by bounding the right side of _____ for particular choices of V _

Convenient bounds are obtained by selecting V to be an interpolant of the exact solution

u_ Bounds furnished in this manner generally provide the exact order of convergence in

the mesh spacing h_ Furthermore_ results similar to _____ may be obtained in other

norms_ They are rarely as precise as those in strain energy and typically indicate that

the _nite element solution di_ers by no more than a constant from the optimal solution

in the considered norm_

Thus_ we will study the errors associated with interpolation problems_ This can be

done either on a physical or a canonical element_ but we will proceed using a canonical

element since we constructed shape functions in this manner_ For our present purposes_

we regard u__ as a known function that is interpolated by a p th_degree polynomial U__

on the canonical element ____ ___ Any form of the interpolating polynomial may be used_

We use the Lagrange form ______ where

U__ _

Xk__

ckNk__ _____

with Nk__ given by _____b_ _We have omitted the elemental index e on Nk for clarity

since we are concerned with one element_ An analysis of interpolation errors whith hi_

erarchical shape functions may also be done _cf_ Problem _ at the end of this section_

Although the Lagrangian and hierarchical shape functions di_er_ the resulting interpola_

tion polynomials U__ and their errors are the same since the interpolation problem has

a unique solution __ ___

Selecting p__ distinct points xii _ ____ ___ i _ __ __ _ _ _ _ p_ the interpolation conditions

are

U__i _ u__i __ ui _ ci_ j _ __ __ _ _ _ _ p_ _____

where the rightmost condition follows from _____a_

There are many estimates of pointwise interpolation errors_ Here is a typical result_

Theorem ______ Let u__ _ Cp______ __ then_ for each _ _ ____ ___ there exists a point

___ _ ____ _ such that the error in interpolating u__ by a p th_degree polynomial U__

e__ _

u p____

_p _ _#

Yi__

__ _ _i_ _____

Proof_ Although the proof is not di_cult_ we_ll just sketch the essential details_ A com_

plete analysis is given in numerical analysis texts such as Burden and Faires ___ Chapter

__ and Isaacson and Keller ____ Chapter __

____ Interpolation Errors _

Since

e___ _ e___ _ _ _ _ _ e__p _ _

the error must have the form

e__ _ g__

Yi__

__ _ _i_

The error in interpolating a polynomial of degree p or less is zero_ thus_ g__ must be

proportional to u p___ We may use a Taylor_s series argument to infer the existence of

___ _ ____ _ and

e__ _ Cu p____

Yi__

__ _ _i_

Selecting u to be a polynomial of degree p _ _ and di_erentiating this expression p _ _

times yields C as _ _p _ _# and ______

The pointwise error _____ can be used to obtain a variety of global error estimates_

Let us estimate the error when interpolating a smooth function u__ by a linear polyno_

mial U__ at the vertices __ _ __ and __ _ _ of an element_ Using _____ with p _ _

reveals

e__ _

u____

__ _ ___ _ __ _ _ ____ __ _____

Thus_

je__j _

max

______ ju____j max

______ j__ _ _j_

Now_

max

______ j__ _ _j _ __

Thus_

je__j _

max

______ ju____j_

Derivatives in this expression are taken with respect to __ In most cases_ we would

like results expressed in physical terms_ The linear transformation _____ provides the

necessary conversion from the canonical element to element j_ _xj___ xj__ Thus_

d_u__

d__ _

d_u__

dx_

with hj _ xj _ xj___ Letting

kf__k__j __ max

xj___x_xj jf_xj ______

_ One_Dimensional Finite Element Methods

denote the local _maximum norm_ of f_x on _xj___ xj__ we have

ke__k__j _

j_ ku____k__j_ ______

_Arguments have been replaced by a _ to emphasize that the actual norm doesn_t depend

on x_

If u_x were interpolated by a piecewise_linear function U_x on N elements _xj___ xj __

j _ __ _ _ _ _ _N_ then ______ could be used on each element to obtain an estimate of the

maximum error as

ke__k_ _

_ ku____k__ _____a

where

kf__k_ __ max

__j_N kf__k__j_ _____b

and

h __ max

__j_N

_xj _ xj___ _____c

As a next step_ let us use _____ and _____ to compute an error estimate in the L_

norm_ thus_

Z xj

xj__

e__xdx _

Z _

u______

___ _ ___d__

Since j__ _ _j _ __ we have

Z xj

xj__

e__xdx _

_ Z _

_u________d__

Introduce the _local L_ norm_ of a function f_x as

kf__k__j __ _Z xj

xj__

f__xdx____

_ ______

Then_

ke__k_

_j _

_ Z _

_u________d__

It is tempting to replace the integral on the right side of our error estimate by ku__k_

_j _

This is almost correct_ however_ _ _ ____ We would have to verify that _ varies smoothly

with __ Here_ we will assume this to be the case and expand u__ using Taylor_s theorem

to obtain

u____ _ u____ _ u_______ _ _ _ u____ _ O_j_ _ _j_ _ _ ___ __

____ Interpolation Errors _

ju____j _ Cju____j_

The constant C absorbs our careless treatment of the higher_order term in the Taylor_s

expansion_ Thus_ using ______ we have

ke__k_

_j _ C_ hj

_ Z _

_u______d_ _ C_ h_

j __ Z xj

xj__

_u___x__dx_

where derivatives in the rightmost expression are with respect to x_ Using ______

ke__k_

_j _ C_ h_

j __ku____k_

_j _ ______

If we sum ______ over the N _nite elements of the mesh and take a square root we

obtain

ke__k_ _ Ch_ku____k__ ______a

where

kf__k_

Xj__

kf__k_

_j _ ______b

_The constant C in ______a replaces the constant C _ of _______ but we won_t be

precise about identifying di_erent constants_

With a goal of estimating the error in H__ let us examine the error u___ _ U____

Di_erentiating _____ with respect to _

e___ _ u_____ _

u_____

___ _ __

Assuming that d_ d_ is bounded_ we use ______ and _____ to obtain

ke_k_

_j _ Z xj

xj__

de_x

__dx _

hj Z _

_u_____ _

u_____

___ _ ___d__

Following the arguments that led to _______ we _nd

ke___k_

_j _ Ch_

j ku____k_

_j _

Summing over the N elements

ke___k_

_ _ Ch_ku____k__ ______

__ One_Dimensional Finite Element Methods

To obtain an error estimate in the H_ norm_ we combine ______a and ______ to get

ke__k_ _ Chku____k_ ______a

where

kf__k_

Xj__

_kf___k_

_j _ kf__k_

_j__ ______b

The methodology developed above may be applied to estimate interpolation errors of

higher_degree polynomial approximations_ A typical result follows_

Theorem ______ Introduce a mesh a _ x_ _ x_ _ _ _ _ _ xN _ b such that U_x is a

polynomial of degree p or less on every subinterval _xj___ xj and U _ H__a_ b_ Let U_x

interpolate u_x _ Hp___a_ b_ such that no error results when u_x is any polynomial of

degree p or less_ Then_ there exists a constant Cp _ __ depending on p_ such that

ku _ Uk_ _ Cphp__ku p__k_ ______a

and

ku _ Uk_ _ Chpp

ku p__k__ ______b

where h satis_es _____ _c__

Proof_ The analysis follows the one used for linear polynomials_

Problems

__ Choose a hierarchical polynomial _____ on a canonical element ____ __ and show

how to determine the coe_cients cj_ j _ ___ __ _ _ _ _ _ p_ to solve the interpolation

problem ______

Bibliography

___ M_ Abromowitz and I_A_ Stegun_ Handbook of Mathematical Functions_ volume __ of

Applied Mathematics Series_ National Bureau of Standards_ Gathersburg_ _____

__ R_L_ Burden and J_D_ Faires_ Numerical Analysis_ PWS_Kent_ Boston_ _fth edition_

_____

___ G_F_ Carey and J_T_ Oden_ Finite Elements A Second Course_ volume II_ Prentice

Hall_ Englewood Cli_s_ _____

___ R_ Courant and D_ Hilbert_ Methods of Mathematical Physics_ volume __ Wiley_

Interscience_ New York_ _____

___ C_ de Boor_ A Practical Guide to Splines_ Springer_Verlag_ New York_ _____

___ E_ Isaacson and H_B_ Keller_ Analysis of Numerical Methods_ John Wiley and Sons_

New York_ _____

___ B_ Szab o and I_ Babu!ska_ Finite Element Analysis_ John Wiley and Sons_ New York_

_____

Chapter _

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