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Multi_Dimensional Variational Principles

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___ Galerkin_s Method and Extremal Principles

The construction of Galerkin formulations presented in Chapters _ and _ for one_dimensional

problems readily extends to higher dimensions_ Following our prior developments_ we_ll

focus on the model two_dimensional self_adjoint di_usion problem

L_u __p_x_ y_ux_x _ _p_x_ y_uy_y q_x_ y_u f_x_ y__ _x_ y_ _ __ ______a_

where _ _ __ with boundary __ _Figure ______ and p_x_ y_ _ __ q_x_ y_ _ __ _x_ y_ _ __

Essential boundary conditions

u_x_ y_ __x_ y__ _x_ y_ _ __E_ ______b_

are prescribed on the portion __E of __ and natural boundary conditions

p_x_ y_

_u_x_ y_

pru _ n _ p_ux cos _ uy sin __ __x_ y__ _x_ y_ _ __N_

______c_

are prescribed on the remaining portion __N of ___ The angle _ is the angle between

the x_axis and the outward normal n to __ _Figure _______

The Galerkin form of _______ is obtained by multiplying ______a_ by a test function v

and integrating over _ to obtain

v___pux_x _ _puy_y qu _ f dxdy __ _______

In order to integrate the second derivative terms by parts in two and three dimensions_

we use Green_s theorem or the divergence theorem

r _ adxdy Z

a _ nds ______a_

_ Multi_Dimensional Variational Principles

s n

pu = n





u = 

Figure ______ Two_dimensional region _ with boundary __ and normal vector n to ___

where s is a coordinate on ___ a _a__ a_ T_ and

r _ a

_a_

_ ______b_

In order to use this result in the present circumstances_ let us introduce vector notation

_pux_x _puy_y _ r _ _pru_

and use the _product rule_ for the divergence and gradient operators

r _ _vpru_ _rv_ _ _pru_ vr _ _pru__ ______c_

Thus_

_vr _ _pru_dxdy ZZ

__rv_ _ _pru__r_ _vpru_ dxdy_

Now apply the divergence theorem _______ to the second term to obtain

_vr _ _pru_dxdy ZZ

rv _ prudxdy _ Z

vpru _ nds_

Thus_ _______ becomes

_rv _ pru v_qu _ f_ dxdy _ Z

vpunds _ _______

____ Galerkin_s Method and Extremal Principles _

where ______c_ was used to simplify the surface integral_

The integrals in _______ must exist and_ with u and v of the same class and p and q

smooth_ this implies

_u_

x u_

y u__dxdy

exists_ This is the two_dimensional Sobolev space H__ Drawing upon our experiences

in one dimension_ we expect u _ H_

E_ where functions in H_

E are in H_ and satisfy the

Dirichlet boundary conditions ______b_ on _E_ Likewise_ we expect v _ H_

_ _ which denotes

that v _ on __E_ Thus_ the variation v should vanish where the trial function u is

prescribed_

Let us extend the one_dimensional notation as well_ Thus_ the L_ inner product is

_v_ f_ _ ZZ

vfdxdy ______a_

and the strain energy is

A_v_ u_ _ _rv_ pru_ _v_ qu_ ZZ

_p_vxux vyuy_ qvu dxdy_ ______b_

We also introduce a boundary L_ inner product as

_ v_w _ Z

__N

vwds_ ______c_

The boundary integral may be restricted to __N since v _ on __E_ With this nomen_

clature_ the variational problem _______ may be stated as_ _nd u _ H_

E satisfying

A_v_ u_ _v_ f_ _ v__ __ _v _ H_

_ _ _______

The Neumann boundary condition ______c_ was used to replace pun in the boundary

inner product_ The variational problem _______ has the same form as the one_dimensional

problem ________ Indeed_ the theory and extremal principles developed in Chapter _ apply

to multi_dimensional problems of this form_

Theorem ______ The function w _ H_

E that minimizes

I_w A_w_w_ _ __w_ f_ _ _ _ w__ _ _ _______

is the one that satis_es ________ and conversely_

Proof_ The proof is similar to that of Theorem _____ and appears as Problem _ at the

end of this section_

_ Multi_Dimensional Variational Principles

Corollary ______ Smooth functions u _ H_

E satisfying _______ or minimizing _____ _ also

satisfy ________

Proof_ Again_ the proof is left as an exercise_

Example ______ Suppose that the Neumann boundary conditions ______c_ are changed

to Robin boundary conditions

pun u __ _x_ y_ _ __N_ ______a_

Very little changes in the variational statement of the problem ______a_b__ ________ Instead

of replacing pun by _ in the boundary inner product ______c__ we replace it by _ _ u_

Thus_ the Galerkin form of the problem is_ _nd u _ H_

E satisfying

A_v_ u_ _v_ f_ _ v__ _ u __ _v _ H_

_ _ ______b_

Example _____ Variational principles for nonlinear problems and vector systems

of partial di_erential equations are constructed in the same manner as for the linear

scalar problems ________ As an example_ consider a thin elastic sheet occupying a two_

dimensional region __ As shown in Figure ______ the Cartesian components _u__ u__ of

the displacement vector vanish on the portion __E of of the boundary __ and the com_

ponents of the traction are prescribed as _S__ S__ on the remaining portion __N of ___

The equations of equilibrium for such a problem are _cf__ e_g__ __ _ Chapter __

___

__ ______a_

___

__ _x_ y_ _ __ ______b_

where ij _ i_ j __ __ are the components of the two_dimensional symmetric stress tensor

_matrix__ The stress components are related to the displacement components by Hooke_s

law

_ _ __ _

_u_

__ _______a_

_ _ __ __

_u_

__ _______b_

___ __

_u_

__ _______c_

____ Galerkin_s Method and Extremal Principles _

s n



u = 0

u = 0,



Figure ______ Two_dimensional elastic sheet occupying the region __ Displacement com_

ponents _u__ u__ vanish on __E and traction components _S__ S__ are prescribed on __N_

where E and _ are constants called Young_s modulus and Poisson_s ratio_ respectively_

The displacement and traction boundary conditions are

u__x_ y_ __ u__x_ y_ __ _x_ y_ _ __E_ _______a_

n___ n___ S__ n___ n___ S__ _x_ y_ _ __N_ _______b_

where n _n__ n_ T _cos __ sin _ T is the unit outward normal vector to __ _Figure

_______

Following the one_dimensional formulations_ the Galerkin form of this problem is

obtained by multiplying ______a_ and ______b_ by test functions v_ and v__ respectively_

integrated over __ and using the divergence theorem_ With u_ and u_ being components

of a displacement _eld_ the functions v_ and v_ are referred to as components of the

virtual displacement _eld_

We use ______a_ to illustrate the process_ thus_ multiplying by v_ and integrating over

__ we _nd

v__

___

dxdy __

The three stress components are dependent on the two displacement components and

are typically replaced by these using _________ Were this done_ the variational principle

_ Multi_Dimensional Variational Principles

would involve second derivatives of u_ and u__ Hence_ we would want to use the divergence

theorem to obtain a symmetric variational form and reduce the continuity requirements

on u_ and u__ We_ll do this_ but omit the explicit substitution of ________ to simplify the

presentation_ Thus_ we regard __ and __ as components of a two_vector_ we use the

divergence theorem _______ to obrain

_v_

__ dxdy Z

v__n___ n___ ds_

Selecting v_ _ H_

_ implies that the boundary integral vanishes on __E_ This and the

subsequent use of the natural boundary condition _______b_ give

_v_

__ dxdy Z

__N

v_S_ds_ _v_ _ H_

_ _ _______a_

Similar treatment of ______b_ gives

_v_

__ dxdy Z

__N

v_S_ds_ _v_ _ H_

_ _ _______b_

Equations _______a_ and _______b_ may be combined and written in a vector form_

Letting u _u__ u_ T _ etc__ we add _______a_ and _______b_ to obtain the Galerkin problem_

_nd u _ H_

_ such that

A_v_ u_ _ v_ S __ _v _ H_

_ _ _______a_

where

A_v_ u_ ZZ

_v_

__ _

_v_

___ dxdy_ _______b_

_ v_ S _ Z

__N

_v_S_ v_S__ds_ _______c_

When a vector function belongs to H__ we mean that each of its components is in H__

The spaces H_

E and H_

_ are identical since the displacement is trivial on __E_

The solution of ________ also satis_es the following minimum problem_

Theorem ______ Among all functions w _w__ w_ T _ H_

E the solution u _u__ u_ T of

________ is the one that minimizes

I_w

___ _ ___ ZZ

f__ _ ____

_w_

__ _

_w_

__ __

_w_

____ Galerkin_s Method and Extremal Principles _

__ _ __

_w_

__gdxdy _ Z

__N

_w_S_ w_S__ds_

and conversely_

Proof_ The proof is similar to that of Theorem ______ The stress components ij _ i_ j

__ __ have been eliminated in favor of the displacements using _________

Let us conclude this section with a brief summary_

_ A solution of the di_erential problem_ e_g__ ________ is called a _classical_ or _strong_

solution_ The function u _ H_

B_ where functions in H_ have _nite values of

__uxx__ _uxy__ _uyy__ _ux__ _uy__ u_ dxdy

and functions in H_

B also satisfy all prescribed boundary conditions_ e_g__ ______b_c__

_ Solutions of a Galerkin problem such as _______ are called _weak_ solutions_ They

may be elements of a larger class of functions than strong solutions since the high_

order derivatives are missing from the variational statement of the problem_ For

the second_order di_erential equations that we have been studying_ the variational

form _e_g__ ________ only contains _rst derivatives and u _ H_

E_ Functions in H_

have _nite values of

__ux__ _uy__ u_ dxdy_

and functions in H_

E also satisfy the prescribed essential _Dirichlet_ boundary con_

dition ______b__ Test functions v are not varied where essential data is prescribed

and are elements of H_

_ _ They satisfy trivial versions of the essential boundary

conditions_

_ While essential boundary conditions constrain the trial and test spaces_ natural

_Neumann or Robin_ boundary conditions alter the variational statement of the

problem_ As with _______ and _________ inhomogeneous conditions add boundary

inner product terms to the variational statement_

_ Smooth solutions of the Galerkin problem satisfy the original partial di_erential

equation_s_ and natural boundary conditions_ and conversely_

_ Galerkin problems arising from self_adjoint di_erential equations also satisfy ex_

tremal problems_ In this case_ approximate solutions found by Galerkin_s method

are best in the sense of ________ i_e__ in the sense of minimizing the strain energy of

the error_

_ Multi_Dimensional Variational Principles

Problems

__ Prove Theorem _____ and its Corollary_

__ Prove Theorem _____ and aslo show that smooth solutions of ________ satisfy the

di_erential system _______ _ _________

__ Consider an in_nite solid medium of material M containing an in_nite number of

periodically spaced circular cylindrical _bers made of material F_ The _bers are

arranged in a square array with centers two units apart in the x and y directions

_Figure _______ The radius of each _ber is a __ ___ The aim of this problem is to

_nd a Galerkin problem that can be used to determine the e_ective conductivity

of the composite medium_ Because of embedded symmetries_ it su_ces to solve a

F a



Figure ______ Composite medium consisting of a regular array of circular cylindrical _bers

embedded in in a matrix _left__ Quadrant of a Periodicity cell used to solve this problem

_right__

problem on one quarter of a periodicity cell as shown on the right of Figure ______

The governing di_erential equations and boundary conditions for the temperature

____ Galerkin_s Method and Extremal Principles _

_or potential_ etc__ u_x_ y_ within this quadrant are

r _ _pru_ __ _x_ y_ _ _F _M_

ux___ y_ ux___ y_ __ _ y __

u_x_ __ __ u_x_ __ __ _ x __

u _ C__ pur _ C__ _x_ y_ _ x_ y_ a__

________

The subscripts F and M are used to indicate the regions and properties of the _ber

and matrix_ respectively_ Thus_ letting

_ _ f_x_ y_j _ x __ _ y _g_

we have

_F _ f_r_ __j _ r a_ _ _ __g_

and

_M _ _ _ _F _

The conductivity p of the _ber and matrix will generally be di_erent and_ hence_ p

will jump at r a_ If necessary_ we can write

p_x_ y_ _ pF _ if x_ y_ _ a_

pM_ if x_ y_ _ a_ _

Although the conductivities are discontinuous_ the last boundary condition con_rms

that the temperature u and _ux pur are continuous at r a_

____ Following the steps leading to ________ show that the Galerkin form of this

problem consists of determining u _ H_

E as the solution of

_F __M

p_uxvx uyvy_dxdy __ _v _ H_

_ _

De_ne the spaces H_

E and H_

_ for this problem_ The Galerkin problem appears

to be the same as it would for a homogeneous medium_ There is no indication

of the continuity conditions at r a_

____ Show that the function w _ H_

E that minimizes

I_w ZZ

_F__M

p_w_

x w_

y_dxdy

is the solution u of the Galerkin problem_ and conversely_ Again_ there is little

evidence that the problem involves an inhomogeneous medium_

__ Multi_Dimensional Variational Principles

___ Function Spaces and Approximation

Let us try to formalize some of the considerations that were raised about the properties

of function spaces and their smoothness requirements_ Consider a Galerkin problem in

the form of ________ Using Galerkin_s method_ we _nd approximate solutions by solving

_______ in a _nite_dimensional subspace SN of H__ Selecting a basis f_jgNj

__ for SN_ we

consider approximations U _ SN

E of u in the form

U_x_ y_

Xj__

cj_j_x_ y__ _______

With approximations V _ SN

_ of v having a similar form_ we determine U as the solution

A_V_U_ _V_ f_ _ V__ __ _V _ SN

_ _ _______

_Nontrivial essential boundary conditions introduce di_erences between SN

E and SN

_ and

we have not explicitly identi_ed these di_erences in _________

We_ve mentioned the criticality of knowing the minimum smoothness requirements

of an approximating space SN_ Smooth _e_g_ C__ approximations are di_cult to con_

struct on nonuniform two_ and three_dimensional meshes_ We have already seen that

smoothness requirements of the solutions of partial di_erential equations are usually ex_

pressed in terms of Sobolev spaces_ so let us de_ne these spaces and examine some of

their properties_ First_ let_s review some preliminaries from linear algebra and functional

analysis_

De_nition ______ V is a linear space if

__ u_ v _ V then u v _ V_

__ u _ V then _u _ V_ for all constants __ and

__ u_ v _ V then _u _v _ V_ for all constants __ __

De_nition ______ A_u_ v_ is a bilinear form on V_V if_ for u_ v_w _ V and all constants

_ and __

__ A_u_ v_ _ __ and

__ A_u_ v_ is linear in each argument_ thus_

A_u_ _v _w_ _A_u_ v_ _A_u_w__

A__u _v_w_ _A_u_w_ _A_v_w__

____ Function Spaces and Approximation __

De_nition ______ An inner product A_u_ v_ is a bilinear form on V _V that

__ is symmetric in the sense that A_u_ v_ A_v_ u__ _u_ v _ V_ and

__ A_u_ u_ _ __ u _ _ and A_____ __ _u _ V_

De_nition ______ The norm k _ kA associated with the inner product A_u_ v_ is

kukA pA_u_ u_ _______

and it satis_es

__ kukA _ __ u _ __ k_kA __

__ ku vkA kukA kvkA_ and

__ k_ukA j_jkukA_ for all constants __

The integrals involved in the norms and inner products are Lebesgue integrals rather

than the customary Riemann integrals_ Functions that are Riemann integrable are also

Lebesgue integrable but not conversely_ We have neither time nor space to delve into

Lebesgue integration nor will it be necessary for most of our discussions_ It is_ however_

helpful when seeking understanding of the continuity requirements of the various function

spaces_ So_ we_ll make a few brief remarks and refer those seeking more information to

texts on functional analysis ___ __ _ _

With Lebesgue integration_ the concept of the length of a subinterval is replaced by

the measure of an arbitrary point set_ Certain sets are so sparse as to have measure

zero_ An example is the set of rational numbers on ___ _ _ Indeed_ all countably in_nite

sets have measure zero_ If a function u _ V possesses a given property except on a set

of measure zero then it is said to have that property almost everywhere_ A relevant

property is the notion of an equivalence class_ Two functions u_ v _ V belong to the same

equivalence class if

ku _ vkA __

With Lebesgue integration_ two functions in the same equivalence class are equal almost

everywhere_ Thus_ if we are given a function u _ V and change its values on a set of

measure zero to obtain a function v_ then u and v belong to the same equivalence class_

We need one more concept_ the notion of completeness_ A Cauchy sequence fung_

__ _

V is one where

lim

m_n__

kum _ unkA __

__ Multi_Dimensional Variational Principles

If fung_

__ converges in k _ kA to a function u _ V then it is a Cauchy sequence_ Thus_

using the triangular inequality_

lim

m_n__

kum _ unkA lim

m_n__

fkum _ ukA ku _ unkAg __

A space V where the converse is true_ i_e__ where all Cauchy sequences fung_

__ converge

in k _ kA to functions u _ V_ is said to be complete_

De_nition ______ A complete linear space V with inner product A_u_ v_ and correspond_

ing norm kukA_ u_ v _ V is called a Hilbert space_

Let_s list some relevant Hilbert spaces for use with variational formulations of bound_

ary value problems_ We_ll present their de_nitions in two space dimensions_ Their ex_

tension to one and three dimensions is obvious_

De_nition ______ The space L____ consists of functions satisfying

L____ _ fuj ZZ

u_dxdy _ g_ ______a_

It has the inner product

_u_ v_ ZZ

uvdxdy ______b_

and norm

kuk_ p_u_ u__ ______c_

De_nition ____ _ The Sobolev space Hk consists of functions u which belong to L_ with

their _rst k _ _ derivatives_ The space has the inner product and norm

_u_ v_k _ Xj_j_k

_D_u_D_v__ ______a_

kukk p_u_ u_k_ ______b_

where

_ ____ __ T _ j_j __ ___ ______c_

with __ and __ non_negative integers_ and

D_u _

______u

_x___y__

_ ______d_

____ Function Spaces and Approximation __

In particular_ the space H_ has the inner product and norm

_u_ v__ _u_ v_ _ux_ vx_ _uy_ vy_ ZZ

_uv uxvx uyvy_dxdy ______a_

kuk_ _

_u_ u_

x u_

y_dxdy_

___

_ ______b_

Likewise_ functions u _ H_ have _nite values of

kuk_

_ ZZ

_u_

xx u_

xy u_

yy u_

x u_

y u_ dxdy_

Example _____ We have been studying second_order di_erential equations of the

form _______ and seeking weak solutions u _ H_ and U _ SN _ H_ of _______ and ________

respectively_ Let us verify that H_ is the correct space_ at least in one dimension_ Thus_

consider a basis of the familiar piecewise_linear hat functions on a uniform mesh with

spacing h _N

_j_x_ __

_x _ xj___h_ if xj__ x _ xj

_xj__ _ x_h_ if xj x _ xj__

__ otherwise

_ _______

Since SN _ H__ _j and __

j must be in L__ j __ __ ____N_ Consider C_ approximations of

_j_x_ and __

j_x_ obtained by _rounding corners_ in O_hn__neighborhoods of the nodes

xj___ xj _ xj__ as shown in Figure______ A possible smooth approximation of __

j_x_ is

j_x_ _ __

j_n_x_

_tanh

n_x _ xj___

tanh

n_x _ xj___

_ _ tanh

n_x _ xj_

A smooth approximation _j_n of _j is obtained by integration as

_j_n_x_

ln cosh n__x _ xj___h_ cosh n__x _ xj___h_

cosh_ n__x _ xj_h_ _

Clearly_ _j_n and __

j_n are elements of L__ The _rounding_ disappears as n_ and

lim

n__Z _

___

j_n_x_ _dx _ _h__h__ _h_

The explicit calculations are somewhat involved and will not be shown_ However_ it

seems clear that the limiting function __

j _ L_ and_ hence_ _j _ SN for _xed h_

__ Multi_Dimensional Variational Principles

−1.5 −1 −0.5 0 0.5 1 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

0.8

−1.5 −1 −0.5 0 0.5 1 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure ______ Smooth version of a piecewise linear hat function _______ _top__ its _rst

derivative _center__ and the square of its _rst derivative _bottom__ Results are shown

with xj__ ___ xj __ xj__ _ _h ___ and n ___

Example ____ Consider the piecewise_constant basis function on a uniform mesh

_j_x_ _ __ if xj__ x _ xj

__ otherwise

_ _______

A smooth version of this function and its _rst derivative are shown in Figure _____ and

may be written as

_j_n_x_

_tanh

n_x _ xj___

_ tanh

n_x _ xj_

j_n_x_

_sech_n_x _ xj___

_ sech_ n_x _ xj_

As n _ _ _j_n approaches a square pulse_ however_ __

j_n is proportional to the combi_

nation of delta functions

j_n_x_ _ __x _ xj___ _ __x _ xj__

____ Function Spaces and Approximation __

Thus_ we anticipate problems since delta functions are not elements of L__ Squaring

j_n_x_

___

j_n_x_ _ _

___sech_n_x _ xj___

__sech_n_x _ xj___

sech_n_x _ xj_

sech_ n_x _ xj_

As shown in Figure ______ the function sechn_x _ xj_h is largest at xj and decays

exponentially fast from xj_ thus_ the center term in the above expression is exponentially

small relative to the _rst and third terms_ Neglecting it yields

___

j_n_x_ _ _ _

___sech_n_x _ xj___

sech_n_x _ xj_

Thus_

Z _

___

j_n_x_ _dx _

__h

_tanh

n_x _ xj___

__ sech_ n_x _ xj___

tanh

n_x _ xj_

__ sech_ n_x _ xj_

_ _

This is unbounded as n__ hence_ __

j_x_ _ L_ and _j_x_ _ H__

−0.5 0 0.5 1 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.5 0 0.5 1 1.5

−10

−8

−6

−4

−2

Figure ______ Smooth version of a piecewise constant function _______ _left_ and its _rst

derivative _right__ Results are shown with xj__ __ xj _ _h ___ and n ___

Although the previous examples lack rigor_ we may conclude that a basis of continuous

functions will belong to H_ in one dimension_ More generally_ u _ Hk implies that

u _ Ck__ in one dimension_ The situation is not as simple in two and three dimensions_

The Sobolev space Hk is the completion with respect to the norm _______ of Ck functions

whose _rst k partial derivatives are elements of L__ Thus_ for example_ u _ H_ implies

that u_ ux_ and uy are all elements of L__ This is not su_cient to ensure that u is

continuous in two and three dimensions_ Typically_ if __ is smooth then u _ Hk implies

that u _ Cs__ ___ where s is the largest integer less than _k _ d__ in d dimensions

___ _ _ In two and three dimensions_ this condition implies that u _ Ck___

Problems

__ Multi_Dimensional Variational Principles

__ Assuming that p_x_ y_ _ _ and q_x_ y_ _ __ _x_ y_ _ __ _nd any other conditions

that must be satis_ed for the strain energy

A_v_ u_ ZZ

_p_vxux vyuy_ qvu dxdy

to be an inner product and norm_ i_e__ to satisfy De_nitions _____ and ______

__ Construct a variational problem for the fourth_order biharmonic equation

__p_u_ f_x_ y__ _x_ y_ _ __

where

_u uxx uyy

and p_x_ y_ _ _ is smooth_ Assume that u satis_es the essential boundary conditions

u_x_ y_ __ un_x_ y_ __ _x_ y_ _ ___

where n is a unit outward normal vector to ___ To what function space should the

weak solution of the variational problem belong_

___ Overview of the Finite Element Method

Let us conclude this chapter with a brief summary of the key steps in constructing a _nite_

element solution in two or three dimensions_ Although not necessary_ we will continue

to focus on _______ as a model_

__ Construct a variational form of the problem_ Generally_ we will use Galerkin_s

method to construct a variational problem_ As described_ this involves multiplying the

di_erential equation be a suitable test function and using the divergence theorem to get

a symmetric formulation_ The trial function u _ H_

E and_ hence_ satis_es any prescribed

essential boundary conditions_ The test function v _ H_

_ and_ hence_ vanishes where

essential boundary conditions are prescribed_ Any prescribed Neumann or Robin bound_

ary conditions are used to alter the variational problem as_ e_g__ with _______ or ______b__

respectively_

Nontrivial essential boundary conditions introduce di_erences in the spaces H_

E and

_ _ Furthermore_ the _nite element subspace SN

E cannot satisfy non_polynomial bound_

ary conditions_ One way of overcoming this is to transform the di_erential equation to

one having trivial essential boundary conditions _cf_ Problem _ at the end of this sec_

tion__ This approach is di_cult to use when the boundary data is discontinuous or when

the problem is nonlinear_ It is more important for theoretical than for practical reasons_

____ Overview of the Finite Element Method __

The usual approach for handling nontrivial Dirichlet data is to interpolate it by the

_nite element trial function_ Thus_ consider approximations in the usual form

U_x_ y_

Xj__

cj_j_x_ y__ _______

however_ we include basis functions _k for mesh entities _vertices_ edges_ k that are on

__E_ The coe_cients ck associated with these nodes are not varied during the solu_

tion process but_ rather_ are selected to interpolate the boundary data_ Thus_ with a

Lagrangian basis where _k_xj_ yj_ _k_j_ we have

U_xk_ yk_ __xk_ yk_ ck_ _xk_ yk_ _ __E_

The interpolation is more di_cult with hierarchical functions_ but it is manageable _cf_

Section _____ We will have to appraise the e_ect of this interpolation on solution accuracy_

Although the spaces SN

E and SN

_ di_er_ the sti_ness and mass matrices can be made

symmetric for self_adjoint linear problems _cf_ Section _____

A third method of satisfying essential boundary conditions is given as Problem _ at

the end of this section_

_ Discretize the domain_ Divide _ into _nite elements having simple shapes_ such

as triangles or quadrilaterals in two dimensions and tetrahedra and hexahedra in three

dimensions_ This nontrivial task generally introduces errors near ___ Thus_ the problem

is typically solved on a polygonal region _

de_ned by the _nite element mesh _Figure

______ rather than on __ Such errors may be reduced by using _nite elements with curved

sides and!or faces near __ _cf_ Chapter ___ The relative advantages of using fewer curved

elements or a larger number of smaller straight_sided or planar_faced elements will have

to be determined_

__ Generate the element sti_ness and mass matrices and element load vector_ Piece_

wise polynomial approximations U _ SN

E of u and V _ SN

_ of v are chosen_ The approx_

imating spaces SN

E and SN

_ are supposed to be subspaces of H_

E and H_

_ _ respectively_

however_ this may not be the case because of errors introduced in approximating the

essential boundary conditions and!or the domain __ These e_ects will also have to be

appraised _cf_ Section _____ Choosing a basis for SN_ we write U and V in the form of

________

The variational problem is written as a sum of contributions over the elements and

the element sti_ness and mass matrices and load vectors are generated_ For the model

problem _______ this would involve solving

Xe__

_Ae_V_U_ _ _V_ f_e_ _ V__ _e __ _V _ SN

_ _ ______a_

__ Multi_Dimensional Variational Principles

________ _

________

s n



u = 

pu + u = 

5 6

K =

1 2 3 4 5 6 7 8

l =

K , le

Figure ______ Two_dimensional domain _ having boundary __ __E __N with unit

normal n discretized by triangular _nite elements_ Schematic representation of the as_

sembly of the element sti_ness matrix Ke and element load vector le into the global

sti_ness matrix K and load vector l_

where

Ae_V_U_ ZZ

_VxpUx VypUy V qU_dxdy_ ______b_

____ Overview of the Finite Element Method __

_V_ f_e ZZ

V fdxdy_ ______c_

_ V__ _e Z

__e____

V _ds_ ______d_

_e is the domain occupied by element e_ and N is the number of elements in the mesh_

The boundary integral ______d_ is zero unless a portion of __e coincides with the boundary

of the _nite element domain _ __

Galerkin formulations for self_adjoint problems such as _______ lead to minimum prob_

lems in the sense of Theorem ______ Thus_ the _nite element solution is the best solution

in SN in the sense of minimizing the strain energy of the error A_u _ U_ u _ U__ The

strain energy of the error is orthogonal to all functions V in SN

E as illustrated in Figure

_____ for three_vectors_

Figure ______ Subspace SN

E of H_

E illustrating the _best_ approximation property of the

solution of Galerkin_s method_

__ Assemble the global sti_ness and mass matrices and load vector_ The element

sti_ness and mass matrices and load vectors that result from evaluating ______b_d_ are

added directly into global sti_ness and mass matrices and a load vector_ As depicted

in Figure ______ the indices assigned to unknowns associated with mesh entities _vertices

as shown_ determine the correct positions of the elemental matrices and vectors in the

global sti_ness and mass matrices and load vector_

__ Multi_Dimensional Variational Principles

_ Solve the algebraic system_ For linear problems_ the assembly of _______ gives rise

to a system of the form

dT __K M_c _ l _ ______a_

where K and M are the global sti_ness and mass matrices_ l is the global load vector_

cT _c__ c__ ____ cN T _ ______b_

and

dT _d__ d__ ____ dN T _ ______c_

Since ______a_ must be satis_ed for all choices of d_ we must have

_K M_c l_ _______

For the model problem ________ KM will be sparse and positive de_nite_ With proper

treatment of the boundary conditions_ it will also be symmetric _cf_ Chapter ___

Each step in the _nite element solution will be examined in greater detail_ Basis

construction is described in Chapter __ mesh generation and assembly appear in Chapter

__ error analysis is discussed in Chapter __ and linear algebraic solution strategies are

presented in Chapter ___

Problems

__ By introducing the transformation

"u u _ _

show that _______ can be changed to a problem with homogeneous essential bound_

ary conditions_ Thus_ we can seek "u _ H_

_ _

__ Another method of treating essential boundary conditions is to remove them by

using a _penalty function__ Penalty methods are rarely used for this purpose_ but

they are important for other reasons_ This problem will introduce the concept and

reinforce the material of Section ____ Consider the variational statement _______ as

an example_ and modify it by including the essential boundary conditions

A_v_ u_ _v_ f_ _ v__ ___N _ _ v__ _u ___E_ _v _ H__

Here _ is a penalty parameter and subscripts on the boundary integral indicate

their domain_ No boundary conditions are applied and the problem is solved for u

and v ranging over the whole of H__

____ Overview of the Finite Element Method __

Show that smooth solutions of this variational problem satisfy the di_erential equa_

tion ______a_ as well as the natural boundary conditions ______c_ and

__ _x_ y_ _ _E_

The penalty parameter _ must be selected large enough for this natural boundary

condition to approximate the prescribed essential condition ______b__ This can be

tricky_ If selected too large_ it will introduce ill_conditioning into the resulting

algebraic system_

__ Multi_Dimensional Variational Principles

Bibliography

__ R_A_ Adams_ Sobolev Spaces_ Academic Press_ New York_ _____

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

Academic Press_ Orlando_ _____

__ C_ Geo_man and G_ Pedrick_ First Course in Functional Analysis_ Prentice_Hall_

Englewood Cli_s_ _____

__ P_R_ Halmos_ Measure Theory_ Springer_Verlag_ New York_ _____

__ J_T_ Oden and L_F_ Demkowicz_ Applied Functional Analysis_ CRC Press_ Boca

Raton_ _____

__ R_ Wait and A_R_ Mitchell_ The Finite Element Analysis and Applications_ John

Wiley and Sons_ Chichester_ _____

Chapter _

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