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Adaptive Finite Element Techniques

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___ Introduction

The usual _nite element analysis would proceed from the selection of a mesh and basis

to the generation of a solution to an accuracy appraisal and analysis_ Experience is the

traditional method of determining whether or not the mesh and basis will be optimal

or even adequate for the analysis at hand_ Accuracy appraisals typically require the

generation of a second solution on a _ner mesh or with a di_erent method and an ad hoc

comparison of the two solutions_ At least with a posteriori error estimation _cf_ Section

_____ accuracy appraisals can accompany solution generation at a lower cost than the

generation of a second solution_

Adaptive procedures try to automatically re_ne_ coarsen_ or relocate a mesh and or

adjust the basis to achieve a solution having a speci_ed accuracy in an optimal fashion_

The computation typically begins with a trial solution generated on a coarse mesh with a

loworder basis_ The error of this solution is appraised_ If it fails to satisfy the prescribed

accuracy_ adjustments are made with the goal of obtaining the desired solution with

minimal e_ort_ For example_ we might try to reduce the discretization error to its desired

level using the fewest degrees of freedom_ While adaptive _nite element methods have

been studied for nearly twenty years ___ __ _ ___ ___ __ ___ ___ ____ surprising little is

known about optimal strategies_ Common procedures studied to date include

_ local re_nement and or coarsening of a mesh _h_re_nement__

_ relocating or moving a mesh _r_re_nement__ and

_ locally varying the polynomial degree of the basis _p_re_nement__

These strategies may be used singly or in combination_ We may guess that rre_nement

alone is generally not capable of _nding a solution with a speci_ed accuracy_ If the mesh

is too coarse_ it might be impossible to achieve a high degree of precision without adding

_ Adaptive Finite Element Techniques

more elements or altering the basis_ Rre_nement is more useful with transient problems

where elements move to follow an evolving phenomena_ By far_ hre_nement is the most

popular ___ ___ ___ __ ___ ____ It can increase the convergence rate_ particularly when

singularities are present _cf_ ___ ___ or Example ______ In some sense pre_nement is the

most powerful_ Exponential convergence rates are possible when solutions are smooth

__ ___ ____ When combined with hre_nement_ these high rates are also possible when

singularities are present ____ ___ ____ The use of pre_nement is most natural with a

hierarchical basis_ since portions of the sti_ness and mass matrices and load vector will

remain unchanged when increasing the polynomial degree of the basis_

A posteriori error estimates provide accuracy appraisals that are necessary to termi

nate an adaptive procedure_ However_ optimal strategies for deciding where and how to

re_ne or move a mesh or to change the basis are rare_ In Section ____ we saw that a pos_

teriori error estimates in a particular norm were computed by summing their elemental

contributions as

kEk_ _

Xe__

kEk_

______

where N_ is the number of elements in the mesh and kEk_

is the restriction of the error

estimate kEk_ to Element e_ The most popular method of determining where adaptivity

is needed is to use kEke as an enrichment indicator_ Thus_ we assume that large errors

come from regions where the local error estimate kEke is large and this is where we should

re_ne or concentrate the mesh and or increase the method order_ Correspondingly_ the

mesh would be coarsened or the polynomial degree of the basis lowered in regions where

kEke is small_ This is the strategy that we_ll follow _cf_ Section ____ however_ we reiterate

that there is no proof of the optimality of enrichment in the vicinity of the largest local

error estimate_

Enrichment indicators other than local error estimates have been tried_ The use of

solution gradients is popular_ This is particularly true of _uid dynamics problems where

error estimates are not readily available ____ ___ ___ ____

In this chapter_ we_ll examine h_ p_ and hpre_nement_ Strategies using rre_nement

will be addressed in Chapter __

___ h_Re_nement

Mesh re_nement strategies for elliptic _steady_ problems need not consider coarsening_

We can re_ne an initially coarse mesh until the requested accuracy is obtained_ This

strategy might not be optimal and won_t be_ for example_ if the coarse mesh is too

_ne in some regions_ Nevertheless_ we_ll concentrate on re_nement at the expense of

____ h_Re_nement _

coarsening_ We_ll also focus on twodimensional problems to avoid the complexities of

threedimensional geometry_

_____ Structured Meshes

Let us _rst consider adaptivity on structured meshes and then examine unstructured

mesh re_nement_ Re_nement of an element of a structured quadrilateralelement mesh

by bisection requires mesh lines running to the boundaries to retain the fourneighbor

structure _cf_ the left of Figure ______ This strategy is simple to implement and has

been used with _nite di_erence computation _____ however_ it clearly re_nes many more

elements than necessary_ The customary way of avoiding the excess re_nement is to

introduce irregular nodes where the edges of a re_ned element meet at the midsides of

a coarser one _cf_ the right of Figure ______ The mesh is no longer structured and our

standard method of basis construction would create discontinuities at the irregular nodes_

Figure _____ Bisection of an element of a structured rectangularelement mesh creating

mesh lines running between the boundaries _left__ The mesh lines are removed by creating

irregular nodes _right__

The usual strategy of handling continuity at irregular nodes is to constrain the basis_

Let us illustrate the technique for a piecewisebilinear basis_ The procedure for higher

order piecewise polynomials is similar_ Thus_ consider an edge between Vertices _ and _

containing an irregular node _ as shown in Figure _____ For simplicity_ assume that the

elements are h _ h squares and that those adjacent to Edge __ are indexed __ __ and _

as shown in the _gure_ For convenience_ let_s also place a Cartesian coordinate system

at Vertex __

We proceed as usual_ constructing shape functions on each element_ Although not

really needed for our present development_ those bilinear shape functions that are nonzero

on Edge __ follow_

_ Adaptive Finite Element Techniques

________

Figure _____ Irregular node at the intersection of a re_ned element_

_ On Element __

N__ _ _

h _ x

__ N__ _ _

h _ x

h _ y

_ On Element __

N__ _ _

h__ _ x

h__

y _ h__

h__

__ N__ _ _

h__ _ x

h__

h _ y

h__

_ On Element __

N__ _ _

h__ _ x

h__

h__ _ y

h__

__ N__ _ _

h__ _ x

h__

As in Chapter __ the second subscript on Nje denotes the element index_

The restriction of U on Element _ to Edge __ is

U_x_ y_ _ c_N___x_ y_ _ c_N___x_ y__

Evaluating this at Node _ yields

U_x__ y__ _

c_ _ c_

_ x _ __

The restriction of U on Elements _ and _ to Edge __ is

U_x_ y_ _ _ c_N___x_ y_ _ c_N___x_ y__ if y _ h__

c_N___x_ y_ _ c_N___x_ y__ if y _ h__

In either case_ we have

U_x__ y__ _ c__ x _ __

Equating the two expressions for U_x__ y__ yields the constraint condition

c_ _

c_ _ c_

_ ______

____ h_Re_nement _

Figure _____ The oneirregular rule_ the intended re_nement of an element to create two

irregular nodes on an edge _left_ necessitates re_nement of a neighboring element to have

no more than one irregular node per element edge _right__

Thus_ instead of determining c_ by Galerkin_s method_ we constrain it to be determined

as the average of the solutions at the two vertices at the ends of the edge_ With the

piecewisebilinear basis used for this illustration_ the solution along an edge containing

an irregular node is a linear function rather than a piecewiselinear function_

Software based on this form of adaptive re_nement has been implemented for elliptic

____ and parabolic ___ systems_ One could guess that di_culties arise when there are too

many irregular nodes on an edge_ To overcome this_ software developers typically use

Bank_s ___ ___ _oneirregular_ and _threeneighbor_ rules_ The oneirregular rule limits

the number of irregular nodes on an element edge to one_ The impending introduction

of a second irregular node on an edge requires re_nement of a neighboring element as

shown in Figure _____ The threeneighbor rule states that any element having irregular

nodes on three of its four edges must be re_ned_

A modi_ed quadtree _Section ____ can be used to store the mesh and solution data_

Thus_ let the root of a tree structure denote the original domain __ With a structured

grid_ we_ll assume that _ is square_ although it could be obtained by a mapping of a

distorted region to a square _Section _____ The elements of the original mesh are regarded

as o_spring of the root _Figure ______ Elements introduced by adaptive re_nement are

obtained by bisection and are regarded as o_spring of the elements of the original mesh_

This structure is depicted in Figure _____ Coarsening can be done by _pruning_ re_ned

quadrants_ It_s customary_ but not essential_ to assume that elements cannot be removed

_by coarsening_ from the original mesh ____

Irregular nodes can be avoided by using transition elements as shown in Figure _____

The strategy on the right uses triangular elements as a transition between the coarse and

_ne elements_ If triangular elements are not desirable_ the transition element on the left

uses rectangles but only adds a midedge shape functions at Node __ There is no node

at the midpoint of Edge ___ The shape functions on the transition element are

N__ _ _

h _ x

y _ h__

h__

__ N__ _ _

h _ x

h__ _ y

h__

_ Adaptive Finite Element Techniques

Figure _____ Original structured mesh and the bisection of two elements _left__ The tree

structure used to represent this mesh _right__

________

_ _

Figure _____ Transition elements between coarse and _ne elements using rectangles _left_

and triangles _right__

N__ _ _

h _ x

__ _ y

h__ __ if _ _ y _ h__

_h_y

h__ __ if h__ _ y _ h

N__ _ __x

__ N__ _ __x

h _ y

Again_ the origin of the coordinate system is at Node __ Those shape functions associated

with nodes on the right edge are piecewisebilinear on Element __ whereas those associated

with nodes on the left edge are linear_

Berger and Oliger ____ considered structured meshes with structured mesh re_nement_

but allowed elements of _ner meshes to overlap those of coarser ones _Figure ______ This

method has principally used with adaptive _nite di_erence computation_ but it has had

some use with _nite element methods _____

_____ Unstructured Meshes

Computation with triangularelement meshes has been done since the beginning of adap

tive methods_ Bank ___ ___ developed the _rst software system PLTMG_ which solves

____ h_Re_nement _

Figure _____ Composite grid construction where _ner grids overlap elements of coarser

ones_

our model problem with a piecewiselinear polynomial basis_ It uses a multigrid itera

tive procedure to solve the resulting linear algebraic system on the sequence of adaptive

meshes_ Bank uses uniform bisection of a triangular element into four smaller elements_

Irregular nodes are eliminated by dividing adjacent triangles sharing a bisected edge

in two _Figure ______ Triangles divided to eliminate irregular nodes are called _green

triangles_ _____ Bank imposes oneirregular and threeneighbor rules relative to green

triangles_ Thus_ e_g__ an intended second bisection of a vertex angle of a green triangle

would not be done_ Instead_ the green triangle would be uniformly re_ned _Figure ____

to keep angles bounded away from zero as the mesh is re_ned_

Figure _____ Uniform bisection of a triangular element into four and the division of

neighboring elements in two _shown dashed__

Rivara ____ ___ developed a mesh re_nement algorithm based on bisecting the longest

edge of an element_ Rivara_s procedure avoids irregular nodes by additional re_nement as

described in the algorithm of Figure _____ In this procedure_ we suppose that elements

Adaptive Finite Element Techniques

Figure ____ Uniform re_nement of green triangles of the mesh shown in Figure ____ to

avoid the second bisection of vertex angles_ New re_nements are shown as dashed lines_

of a submesh _ of mesh _h are scheduled for re_nement_ All elements of _ are bisected

by their longest edges to create a mesh __

h_ which may contain irregular nodes_ Those

elements e of __

h that contain irregular nodes are placed in the set ___ Elements of __ are

bisected by their longest edge to create two triangles_ This bisection may create another

node Q that is di_erent from the original irregular node P of element e_ If so_ P and Q

are joined to produce another element _Figure _______ The process is continued until all

irregular nodes are removed_

procedure rivara__h_ __

Obtain __

h by bisecting all triangles of _ by their longest edges

Let __ contain those elements of __

h having irregular nodes

i __ _

while _i is not _ do

Let e _ _i have an irregular node P and bisect e by its longest edge

Let Q be the intersection point of this bisection

if P __ Q then

Join P and Q

end if

Let _i__

h be the mesh created by this process

Let _i__ be the set of elements in _i__

h with irregular nodes

i __ i _ _

end while

return _i

Figure _____ Rivara_s mesh bisection algorithm_

Rivara_s ____ algorithm has been proven to terminate with a regular mesh in a _nite

number of steps_ It also keep angles bounded away from _ and __ In fact_ if is the

____ h_Re_nement _

P P

Figure ______ Elimination of an irregular node P _left_ as part of Rivara_s algorithm

shown in Figure ____ by dividing the longest edge of Element e and connecting vertices

as indicated_

smallest angle of any triangle in the originalmesh_ the smallest angle in the mesh obtained

after an arbitrary number of applications of the algorithm of Figure _____ is no smaller

than __ _____ Similar procedures were developed by Sewell ____ and used by Mitchell

___ by dividing the newest vertex of a triangle_

Tree structures can be used to represent the data associated with Bank_s ____ and

Rivara_s ____ procedures_ As with structuredmesh computation_ elements introduced

by re_nement are regarded as o_spring of coarser parent elements_ The actual data

representations vary somewhat from the tree described earlier _Figure _____ and readers

seeking more detail should consult Bank ____ or Rivara ____ ____ With tree structures_ any

coarsening may be done by pruning _leaf_ elements from the tree_ Thus_ those elements

nested within a coarser parent are removed and the parent is restored as the element_

As mentioned earlier_ coarsening beyond the original mesh is not allowed_ The process

is complex_ It must be done without introducing irregular nodes_ Suppose_ for example_

that the quartet of small elements _shown with dashed lines_ in the center of the mesh of

Figure ___ were scheduled for removal_ Their direct removal would create three irregular

nodes on the edges of the parent triangle_ Thus_ we would have to determine if removal

of the elements containing these irregular nodes is justi_ed based on errorindication

information_ If so_ the mesh would be coarsened to the one shown in Figure ______

Notice that the coarsened mesh of Figure _____ di_ers from mesh of Figure ____ that

was re_ned to create the mesh of Figure ____ Hence_ re_nement and coarsening may

not be reversible operations because of their independent treatment of irregular nodes_

Coarsening may be done without a tree structure_ Shephard et al_ ___ describe an

_edge collapsing_ procedure where the vertex at one end of an element edge is _collapsed_

onto the one at the other end_ Ai_a ___ describes a twodimensional variant of this

procedure which we reproduce here_ Let P be the polygonal region composed of the union

of elements sharing Vertex V _Figure _______ Let V__ V__ _ _ _ _ Vk denote the vertices on the

k triangles containing V and suppose that error indicators reveal that these elements may

__ Adaptive Finite Element Techniques

Figure ______ Coarsening of a quartet of elements shown with dashed lines in Figure

___ and the removal of surrounding elements to avoid irregular nodes_

V V

4 3

V4 V3

Figure ______ Coarsening of a polygonal region _left_ by collapsing Vertex V onto V_

_right__

be coarsened_ The strategy of collapsing V onto one of the vertices Vj _ j _ __ __ _ _ _ _ k_ is

done by deleting all edges connected to V and then retriangulating P by connecting Vj

to the other vertices of P _cf_ the right of Figure _______ Vertex V is called the collapsed

vertex and Vj is called the target vertex_

Collapsing has to be evaluated for topological compatibility and geometric validity

before it is performed_ Checking for geometric validity prevents situations like the one

shown in Figure _____ from happening_ A collapse is topologically incompatible when_

e_g__ V is on _ and the target vertex Vj is within __ Assuming that V can be collapsed_

the target vertex is chosen to be the one that maximizes the minimum angle of the

resulting retriangulation of P_ Ai_a ___ does no collapsing when the smallest angle that

would be produced by collapsing is smaller than a prescribed minimum angle_ This might

result in a mesh that is _ner than needed for the speci_ed accuracy_ In this case_ the

minimum angle restriction could be waived when V has been scheduled for coarsening

more than a prescribed number of times_ Suppose that the edges h_e_ h_e_ h_e of an

____ h_Re_nement __

element e are indexed such that h_e _ h_e _ h_e_ then the smallest angle _e of Element

e may be calculated as

sin _e _

_Ae

h_eh_e

where Ae is the area of Element e_

V1 V7

3 V2

Figure ______ A situation where the collapse of Vertex V _left_ creates an invalid mesh

_right__

1

2



1 2

Figure ______ Swapping an edge of a pair of elements _left_ to improve element shape

_right__

The shape of elements containing small or large angles that were created during

re_nement or coarsening may be improved by edge swapping_ This procedure operates on

pairs of triangles __ and __ that share a common edge E_ If Q _ __ ___ edge swapping

occurs deleting Edge E and retriangulating Q by connecting the vertices opposite to

Edge E _Figure _______ Swapping can be regarded as a re_nement of Edge E followed

by a collapsing of this new vertex onto a vertex not on Edge E_ As such_ we recognize

that swapping will have to be checked for mesh validity and topological compatibility_

Of course_ it will also have to provide an improved mesh quality_

_____ Re_nement Criteria

Following the introductory discussion of error estimates in Section ___ we assume the

existence of a set of re_nement indicators _e_ e _ __ __ _ _ _ _N__ which are large where

re_nement is desired and small where coarsening is appropriate_ As noted_ these might

__ Adaptive Finite Element Techniques

be the restriction of a global error estimate to Element e

_ kEk_

______

or an ad hoc re_nement indicator such as the magnitude of the solution gradient on the

element_ In either case_ how do we use this error information to re_ne the mesh_ Perhaps

the simplest approach is to re_ne a _xed percentage of elements having the largest error

indicators_ i_e__ re_ne all elements e satisfying

_e _ _ max

__j_N_

_j _ ______

A typical choice of the parameter _ _ ___ __ is ___

We can be more precise when an error estimate of the form ______ with indicators

given by ______ is available_ Suppose that we have an a priori error estimate of the form

kek _ Chp_ _____a_

After obtaining an a posteriori error estimate kEk on a mesh with spacing h_ we could

compute an estimate of the error constant C as

C kEk

hp _ _____b_

The mesh spacing parameter h may be taken as_ e_g__ the average element size

h _ r A

_____c_

where A is the area of __

Suppose that adaptivity is to be terminated when kEk where is a prescribed

tolerance_ Using _____a__ we would like to construct an enriched mesh with a spacing

parameter h

such that

C h

p _

Using the estimate of C computed by _____b__ we have

h _

kEk___p

_ _____a_

Thus_ using _____c__ an enriched mesh of

_ _

A _

kEk___p

_____b_

____ h_Re_nement __

elements will reduce kEk to approximately _

Having selected an estimate of the number of elements N

_ to be in the enriched

mesh_ we have to decide how to re_ne the current mesh in order to attain the prescribed

tolerence_ We may do this by equidistributing the error over the mesh_ Thus_ we would

like each element of the enriched mesh to have approximately the same error_ Using

_______ this implies that

k E k_

where k E ke is the error indicator of Element e of the enriched mesh_ Using this notion_

we divide the error estimate kEk_

by a factor n so that

kEk_

e n

Thus_ each element of the current mesh is divided into n segments such that

_ _kEke

_ ______

In practice_ n and N_ may be rounded up or increased slightly to provide a measure

of assurance that the error criterion will be satis_ed after the next adaptive solution_

The mesh division process may be implemented by repeated applications of a mesh

re_nement algorithm without solving the partial di_erential equation in between_ Thus_

with bisection ____ ____ the elemental error estimate would be halved on each bisected

element_ Re_nement would then be repeated until ______ is satis_ed_

The error estimation process ______ works with coarsening when n _ __ however_

neighboring elements would have to suggest coarsening as well_

Example _____ Rivara ____ solves Laplace_s equation

uxx _ uyy _ __ _x_ y_ _ __

where _ is a regular hexagon inscribed in a unit circle_ The hexagon is oriented with

one vertex along the positive xaxis with a _crack_ through this vertex for _ _ x _ __

y _ __ Boundary conditions are established to be homogeneous Neumann conditions on

the xaxis below the crack and

u_r_ __ _ r___ sin

everywhere else_ This function is also the exact solution of the problem expressed in a

polar frame eminating from the center of the hexagon_ The solution has a singularity

at the origin due to the _reentrant_ angle of __ at the crack tip and the change in

__ Adaptive Finite Element Techniques

boundary conditions from Dirichlet to Neumann_ The solution was computed with a

piecewiselinear _nite element basis using quasiuniform and adaptive hre_nement_ A

residual error estimation procedure similar to those described in Section ___ was used to

appraise solution accuracy _____ Re_nement followed _______

The results shown in Figure _____ indicate that the uniform mesh is converging as

O_N____ where N is the number of degrees of freedom_ We have seen _Section ____ that

uniform hre_nement converges as

kek_ _ C_hmin_p_q_ _ C_N_min_p_q___ ______

where q _ _ depends on the solution smoothness and_ in two dimensions_ N _ h__ For

linear elliptic problems with geometric singularities_ q _ ___ where _ is the maximum

interior angle on __ For the hexagon with a crack_ the interior angles would be ____

_____ and ___ The latter is the largest angle_ hence_ q _ ____ Thus_ with p _ __

convergence should occur at an O_N_____ rate_ however_ the actual rate is lower _Figure

_______

The adaptive procedure has restored the O_N_____ convergence rate that one would

expect of a problem without singularities_ In general_ optimal adaptive hre_nement will

converge as ___ ___

kek_ _ C_hp _ C_N_p___ _____

___ p_ and hp_Re_nement

With pre_nement_ the mesh is not changed but the order of the _nite element basis is

varied locally over the domain_ As with hre_nement_ we must ensure that the basis

remains continuous at element boundaries_ A situation where second and fourthdegree

hierarchical bases intersect along an edge between two square elements is shown on the

left of Figure _____ The seconddegree approximation _shown at the top left_ consists of a

bilinear shape function at each vertex and a seconddegree correction on each edge_ The

fourthdegree approximation _bottom left_ consists of bilinear shape functions at each

vertex_ second_ third and fourthdegree corrections on each edge_ and a fourthdegree

bubble function associated with the centroid _cf_ Section _____ The maximum degree of

the polynomial associated with a mesh entity is identi_ed on the _gure_ The second and

fourthdegree shape functions would be incompatible _discontinuous_ across the common

edge between the two elements_ This situation can be corrected by constraining the

edge functions to the lowerdegree _two_ basis of the top element as shown in the center

____ p_ and hp_Re_nement __

Figure ______ Solution of Example ____ by uniform ___ and adaptive ___ hre_nement

_____

portion of the _gure or by adding third and fourthorder edge functions to the upper

element as shown on the right of the _gure_ Of the two possibilities_ the addition of the

higher degree functions is the most popular_ Constraining the space to the lowerdegree

polynomial could result in a situation where error criteria satis_ed on the element on the

lower left of Figure ____ would no longer be satis_ed on the element in the lowercenter

portion of the _gure_

Remark __ The incompatibility problem just described would not arise with the

hierarchical data structures de_ned in Section ___ since edge functions are blended onto

all elements containing the edge and_ hence_ would always be continuous_

Szab!o ____ developed a strategy for the adaptive variation of p by constructing error

estimates of solutions with local degrees p_ p___ and p__ on Element e and extrapolating

to get an error estimates for solutions of higher degrees_ With a hierarchical basis_ this

is straightforward when p _ __ One could just use the di_erences between higher and

lowerorder solutions or an error estimation procedure as described in Section ____ When

p _ _ on Element e_ local error estimates of solutions having degrees _ and _ are linearly

extrapolated_ Szabo ____ began by generating piecewiselinear _p _ __ and piecewise

quadratic _p _ __ solutions everywhere and extrapolating the error estimates_ Flaherty

and Moore ____ suggest an alternative when p _ __ They obtain a _lowerorder_ piecewise

__ Adaptive Finite Element Techniques

4 4 4

2 2

4 4 4

4 4

1 1

1 1 1 1

1 1

Figure _____ Second and fourthdegree hierarchical shape functions on two square el

ements are incompatible across the common edge between elements _left__ This can be

corrected by removing the third and fourthdegree edge functions from the lower ele

ment _center_ or by adding third and fourthdegree edge functions to the upper element

_right__ The maximum degree of the shape function associated with a mesh entity is

shown in each case_

constant _p _ __ solution by using the value of the piecewiselinear solution at the center

of Element e_ The di_erence between these two _solutions_ furnishes an error estimate

which_ when used with the error estimate for the piecewiselinear solution_ is linearly

extrapolated to higher values of p_

Having estimates of discretization errors as a function of p on each element_ we can

use a strategy similar to ______ to select a value of p to reduce the error on an element

to its desired level_ Often_ however_ a simpler strategy is used_ As indicated earlier_

the error estimate kEke should be of size _N_ on each element of the mesh_ When

enrichment is indicated_ e_g__ when kEk _ _ we can increase the degree of the polynomial

representation by one on any element e where

_e _ _R

_ _____a_

The parameter _e is an enrichment indicator on Element e_ which may be kEke_ and

_R ____ If coarsening is done_ the degree of the approximation on Element e can be

reduced by one when

_e _ _Che

_____b_

where he is the longest edge of Element e and _C ____

The convergence rate of the p version of the _nite element method is exponential when

the solution has no singularities_ For problems with singularities_ pre_nement converges

____ p_ and hp_Re_nement __

kek _ CN_q ______

where q _ _ depends on the solution smoothness ____ ___ ___ ___ ____ _The parameter

q is intended to be generic and is not necessarily the same as the one appearing in

________ With singularities_ the performance of the p version of the _nite element method

depends on the mesh_ Performance will be better when the mesh has been graded near

the singularity_

This suggests combining h and pre_nement_ Indeed_ when proper mesh re_nement is

combined with an increase of the polynomial degree p_ the convergence rate is exponential

kek _ Ce_q_Nq_ ______

where q_ and q_ are positive constants that depend on the smoothness of the exact solution

and the _nite element mesh_ Generating the correct mesh is crucial and its construction is

only known for model problems ____ ___ ___ ___ ____ Oden et al_ ____ developed a strategy

for hpre_nement that involved taking three solution steps followed by an extrapolation_

Some techniques do not attempt to adjust the mesh and the order at the same time_ but_

rather_ adjust either the mesh or the order_ We_ll illustrate one of these_ but _rst cite

the more explicit version of the error estimate ______ given by Babu"ska and Suri ___

kek_ _ C

hmin_p_q_

pq kukmin_p_q____ ______

The mesh must satisfy the uniformity condition_ the polynomialdegree is uniform_ and

u _ Hq___ In this form_ the constant C is independent of h and p_ This result and the

previous estimates indicate that it is better to increase the polynomial degree when the

solution u is smooth _q is large_ and to reduce h near singularities_ Thus_ a possible

strategy would be to increase p in smooth higherror regions and re_ne the mesh near

singularities_ We_ therefore_ need a method of estimating solution smoothness and Ai_a

___ does this by computing the ratio

_e _ _ _e_p___e_p _ ___ if _e_p _ __ __ _

__ otherwise

______

where p is the polynomial degree on Element e_ An argument has been added to the

error indicator on Element e to emphasize its dependence on the local polynomial degree_

As described in Section ___ __p _ __ can be estimated from the part of U involving the

hierarchal corrections of degree p_ Now

_ If _e _ __ the error estimate is decreasing with increasing polynomial degree_ If

enrichment were indicated on Element e_ pre_nement would be the preferred strat

egy_

_ Adaptive Finite Element Techniques

_ If _e _ _ the recommended strategy would be hre_nement_

Ai_a ___ selects pre_nement if _e _ _ and hre_nement if _e _ __ with _ ____ Adjust

ments have to made when p _ _ ____ Coarsening is done by vertex collapsing when all

elements surrounding a vertex have low errors ____

Example _____ Ai_a ___ solves the nonlinear parabolic partial di_erential equation

ut _ _u___ _ u_ _

uxx _ uyy

_ _x_ y_ _ __ t _ __

with the initial and Dirichlet boundary data de_ned so that the exact solution on the

square _ _ f_x_ y_j_ _ x_y _ _g is

u_x_ y_ t_ _

_ _ ep____x_y_tp____

Although this problem is parabolic_ Ai_a ___ kept the temporal error small so that spatial

errors dominate_

Ai_a ___ solved this problem with _ _ ___ by adaptive h_ p_ and hpre_nement

for a variety of spatial error tolerances_ The initial mesh for hre_nement contained

__ triangular elements and used piecewisequadratic _p _ __ shape functions_ For p

re_nement_ the mesh contained __ triangles with p varying from _ to __ The solution

with adaptive hpre_nement was initiated with __ elements and p _ __ The convergence

history of the three adaptive strategies is reported in Figure _____

The solution with hre_nement appears to be converging at an algebraic rate of ap

proximately N_ ___ which is close to the theoretical rate _cf_ ________ There are no

singularities in this problem and the adaptive p and hpre_nement methods appear to

be converging at exponential rates_

This example and the material in this chapter give an introduction to the essential

ideas of adaptivity and adaptive _nite element analysis_ At this time_ adaptive software

is emerging_ Robust and reliable error estimation procedures are only known for model

problems_ Optimal enrichment strategies are just being discovered for realistic problems_

____ p_ and hp_Re_nement __

101 102 103 104

10−3

10−2

10−1

100

Number Of Degrees Of Freedom

Relative Error In H1 Norm

Figure _____ Errors vs_ the number of degrees of freedom N for Example ____ at t _ ____

using adaptive h_ p and hpre_nement ___ _ and __ respectively__

__ Adaptive Finite Element Techniques

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