Research Collection Working Paper The partition of unity finite element method basic theory and applications Author(s): Melenk, J.M.; Babuška, Ivo Publication Date: 1996 Permanent Link: https://doi.org/10.3929/ethz-a-004284735 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection . For more information please consult the Terms of use . ETH Library
Transcript
Research Collection
Working Paper
The partition of unity finite element methodbasic theory and applications
The paper presents the basic ideas and the mathematical foundation of thepartition of unity �nite element method �PUFEM�� We will show how the PUFEMcan be used to employ the structure of the di�erential equation under considerationto construct e�ective and robust methods� Although the method and its theoryare valid in n dimensions� a detailed and illustrative analysis will be given for a onedimensional model problem� We identify some classes of non�standard problemswhich can pro�t highly from the advantages of the PUFEM and conclude this paperwith some open questions concerning implementational aspects of the PUFEM�
Keywords� Finite element method� meshless �nite element method� robust �nite ele�ment methods� �nite element methods for highly oscillatory solutions
�TICAM� The University of Texas at Austin� Austin� TX ������ USA
� Introduction
The aim of this paper is to present a new method for solving di erential equations�the �partition of unity �nite element method� �PUFEM�� We explain the mathematicalfoundation of the PUFEM and discuss some of its features� The most prominent amongthem are
�� the ability to include a priori knowledge about the local behavior of the solution inthe �nite element space�
� the ability to construct �nite element spaces of any desired regularity �as may beimportant for the solution of higher order equations��
�� the fact that the PUFEM falls into the category of �meshless� methods� a mesh inthe classical sense does not have to be created and thus the complicated meshingprocess is avoided�
�� the fact that the PUFEM can be understood as a generalization of the classical h�p� and hp versions of the �nite element method�
In this paper� we will mostly concentrate on the �rst of these four features� In particular�the one dimensional example of section � illustrates the fact that the PUFEM enablesus to construct �nite element spaces which perform very well in cases where the classical�nite element methods fail or are prohibitively expensive� The success of the PUFEM inthis example is precisely due to the fact that the PUFEM o ers an easy way to includeanalytical information about the problem being solved in the �nite element space� Asimilar example was analyzed in ���� for a problem with a boundary layer� Again� thePUFEM permitted the construction of �nite element spaces which account for the bound�ary layer behavior and thus led to a robust method in the sense that the performance ofthe method is independent of the actual strength of the boundary layer� An applicationof the PUFEM to the Timoshenko beam with hard elastic support can be found in �����The paper is organized as follows� The rest of section � establishes that the two mainingredients of �nite element spaces are local approximation properties and some interele�ment continuity� The PUFEM constructs a global conforming �nite element space outof a set of given local approximation spaces � the precise construction is described insection � Therefore� the PUFEM separates the issues of interelement continuity andlocal approximability and allows us to concentrate on �nding good local approximationspaces for a given problem� In section �� we give a few examples of spaces with goodlocal approximation properties for several di erential equations� A detailed example ofthe PUFEM is presented in section � for a one dimensional model problem with roughcoe�cients� In section � we construct local approximation spaces which re�ect the roughbehavior of the solution and show that they are robust� The numerical example of ���illustrates the robust performance of the PUFEM� We conclude the paper in section� with an application of the PUFEM to the two dimensional Helmholtz equation andidentify some open questions concerning implementational aspects�
�
��� The Finite Element Method
The �nite element method �FEM� for the solution of linear problems can be understoodas follows� The problem is formulated in a weak form
�nd u � X � B�u� v� � F �v� �v � Y ���
where X � Y are Hilbert spaces with norms k � kX � k � kY � B � X � Y �� R is continuousand bilinear� and F � Y �� R is continuous and linear� Of course� in all problemsof interest� the spaces X � Y are in�nite dimensional� In the FEM �nite dimensionalsubspaces Xn � X �called the trial spaces�� Yn � Y �called the test spaces� of dimensionn are chosen and the �nite element approximation uFE is de�ned as the solution of
�nd uFE � Xn � B�uFE� v� � F �v� �v � Yn� ��
In order for the approximations uFE to converge to the exact solution u� the followingtwo conditions are necessary�
� Approximability� u can be approximated well by the subspaces Xn� at least� weneed inffku vkXn j v � Xng � � as n��
� Stability� The bilinear form B �together with the subspaces Xn� Yn� satis�es aninf�sup condition �also known as the BB condition� see �����
In particular� if the stability condition holds� then problem �� has a unique solution uFEwhich satis�es
ku uFEkX � C infv�Xn
ku vkX ���
with a constant C � � independent of u and n� Thus the error of the �nite elementapproximation is � up to the constant C � as small as the error of the best approximantin the space Xn� Therefore� given stability� the performance of the �nite element methodis determined by the approximation properties of the spaces Xn for the approximation ofthe solution u� These observations lead to the problem of constructing spaces Xn whichare conforming �i�e�� Xn � X � and which have good approximation properties for theapproximation of the exact solution u�
��� Local Approximability and Interelement Continuity
Let us now consider some of the classical choices of the trial spaces Xn and see how thecondition to be conforming and the approximation properties are realized� In many ap�plications �e�g�� the heat equation� the elasticity equations in displacement formulation�the space X is a subspace of the Sobolev space H�� We will therefore concentrate onthe classical piecewise polynomial subspaces of H�� In the classical FEM the spaces Xn
are chosen such that they have good local approximation properties and are conforming�more precisely� they are chosen to consist of piecewise polynomials �or mapped poly�nomials� and are continuous across element boundaries� In the h�version of the FEM�the polynomial degree is �xed �typically� p � � and the approximation is realized bydecreasing the mesh size h� If the function u to be approximated is su�ciently smooth
�in Hk� say�� an appropriate interpolant Iu �for example� for p � � and piecewise linearfunctions on triangles� one can choose the nodal interpolant� satis�es an estimate of theform
ku IukH� � Ck�phmin�k���p�jujHk ���
where Ck�p is independent of u and h� We see that the approximation properties of theseclassical h�version type �nite element spaces are good whenever the exact solution isnot �rough�� By �rough� we mean here and in the rest of this paper that either higherderivatives of u are not square integrable �i�e�� the case that k is close to �� or thatthey exist but are very large �i�e�� jujHk is big�� In both cases� the approximation withpiecewise polynomial functions performs very poorly and the mesh size h has to be chosenvery small before a reasonable accuracy is achieved �cf� section � and lemma �����In the p version of the FEM� the mesh is �xed and the local approximation is realized bypolynomials �or mapped polynomials� of increasing degree� Again� continuity across theinterelement boundaries is enforced in order to ensure conformity of the �nite elementspaces� The error estimates typically have the form
ku IukH� � Ckp��k���jujHk� ���
and thus the p version can be expected to work well whenever the exact solution isreasonably smooth� however� the p version exhibits the same de�ciencies as the h versionwhenever the exact solution is rough�In conclusion� the approximation properties of both the h and the p version of the �niteelement method are based on the fact that
�� �local approximability� a smooth function can be approximated locally by polyno�mials� and
� �conformity of the �nite element spaces�interelement continuity� polynomial spacesare big enough to absorb extra constraints of continuity across interelement bound�aries without loosing the approximation properties�
Conversely� any system of functions which have good local approximation properties andcan be constrained to satisfy some interelement continuity leads to a good �nite elementmethod�Let us �rst elaborate the problem of local approximability� There are many systems offunctions which have good local approximation properties� For certain types of equations�one can exploit the structure of the di erential equation to construct spaces of functionswhich can approximate the solution even better than the spaces of polynomials� In sec�tion � we give a few examples of spaces which have very good approximation propertiesfor the solution of Laplace�s equation� the homogeneous Helmholtz equation� and theelasticity equations in two dimensions� For example� for the approximation of harmonicfunctions� it is enough to approximate locally with harmonic polynomials�it is not neces�sary to use the full space of polynomials� In the example of Helmholtz�s equation� we seebelow that local approximation can be done with systems of plane waves or with spacesbased on radial Bessel functions� Finally� in section �� we consider a one dimensionalmodel problem with rough coe�cients and construct spaces of functions �which take into
�
account the rough behavior of the coe�cients of the di erential equation� which havegood local approximation properties for the approximation of the �also rough� solution�In this example� the PUFEM based on these special functions leads to a robust method�i�e�� a method which performs as well as the classical FEM performs for a problem withsmooth coe�cients� This is due to the fact that the special ansatz functions incorporatethe rough behavior of the solution�Let us now turn to the problem of conformity of the �nite element space�interelementcontinuity� We have just seen that it is possible to construct many spaces of functions�typically non�polynomial� which have good local approximation properties for the ap�proximation of a solution u of a di erential equation� In general� it is not possible toenforce conformity� i�e�� interelement continuity� for these non�polynomial local approx�imation spaces� The PUFEM� however� o ers a means to construct a conforming spaceout of any given system of local approximation spaces without sacri�cing the approxi�mation properties� This is done as follows� Let f�ig be a system of overlapping patcheswhich cover the domain � of interest� Let f�ig be a partition of unity subordinate tothe cover� On each patch �i� let Vi � H���i� be a space of functions by which uj�i
can be approximated well� The global �nite element space V is then de�ned byP
i �iVi�Theorem �� below states that the global space V inherits the approximation propertiesof the local spaces Vi� i�e�� the function u can be approximated on � by functions of V aswell as the functions uj�i
can be approximated in the local spaces Vi� Moreover� the spaceV inherits the smoothness of the partition of unity �i� In particular� the smoothness ofthe partition of unity enforces the conformity of the global space V �
��� Potential Applications of the PUFEM
We already mentioned above that one potential �eld of application of the PUFEM areproblems where the classical polynomial based FEM fail� In this category fall problemswhere the solution is rough �or highly oscillatory� and the usual piecewise polynomialspaces cannot resolve the essential features of the solution unless the mesh size h is verysmall or the polynomial degree p is very large� In both cases the computational costs arehigh or even too high for today�s computers� Examples of problems with rough or highlyoscillatory solutions are the elasticity equations for laminated materials� materials withsti eners� or the Helmholtz equation for large wave numbers to mention but a few� Insection � the PUFEM is applied to a problem with rough coe�cients�Problems of singularly perturbed type or problems where the solution exhibits a boundarylayer can also be dealt with very successfully in the framework of the PUFEM� If thesingular behavior of the solution is known� the PUFEM allows us to incorporate thisknowledge directly into the �nite element space� In contrast to this� the classical FEMhas to use very small mesh sizes in order to resolve the singular behavior of the solution�������We mentioned above that the PUFEM falls in the general category of �meshless� meth�ods� This feature might be exploited for certain problems where the usual methodsinvolve frequent remeshing� For example� in the problem of the optimal placement of afastener� the engineer has to try several locations of the fastener� For each run� he has toremesh parts of the domain in order to account for the changed position of the fastener�
�
One could construct a local approximation space which models the fastener and thenchanging the position of the fastener simply means changing the local approximationspaces�
� Mathematical Foundation of the PUFEM
In this section� we present a method of constructing conforming subspaces of H����� Weconstruct �nite element spaces which are subspaces of H���� as an example because oftheir importance in applications� We would like to stress at this point that the methodleads to the construction of smoother spaces �subspaces of Hk� k � �� or subspaces ofSobolev spaces W k�p in a straight forward manner� The main technical notion in theconstruction of the PUFEM spaces is the �M�C�� CG� partition of unity�
De�nition ��� Let � � Rn be an open set� f�ig be an open cover of � satisfying a
pointwise overlap condition
�M � N �x � � cardfi jx � �ig �M�
Let f�ig be a Lipschitz partition of unity subordinate to the cover f�ig satisfying
supp�i � closure��i� �i� ���Xi
�i � on �� � �
k�ikL��Rn� � C�� ��
kr�ikL��Rn� � CG
diam�i� ���
where C�� CG are two constants� Then f�ig is called a �M�C�� CG� partition of unitysubordinate to the cover f�ig� The partition of unity f�ig is said to be of degree m � N�
if f�ig � Cm�Rn�� The covering sets f�ig are called patches�
De�nition ��� Let f�ig be an open cover of � � Rn and let f�ig be a �M�C�� CG�
partition of unity subordinate to f�ig� Let Vi � H���i � �� be given� Then the space
V ��Xi
�iVi � fXi
�ivi j vi � Vig � H����
is called the PUFEM space� The PUFEM space V is said to be of degree m � N ifV � Cm����The spaces Vi are referred to as the local approximation spaces�
Theorem ��� Let � � Rn be given� Let f�ig� f�ig� and fVig be as in de�nitions ����
���� Let u � H���� be the function to be approximated� Assume that the local approxi�mation spaces Vi have the following approximation properties� On each patch �i � �� ucan be approximated by a function vi � Vi such that
ku vikL���i��� � ���i��
kr�u vi�kL���i��� � ���i��
�
Then the functionuap �
Xi
�ivi � V � H����
satis�es
ku uapkL���� �pMC�
�Xi
����i�
����
� ����
kr�u uap�kL���� �pM
�Xi
�CG
diam�i
������i� ! C�
�����i�
����
� ����
Proof� We will only show estimate ���� because ���� is proved similarly� Let uap bede�ned as in the statement of the theorem� Since the functions �i form a partition ofunity� we have � � u � �
Pi �i�u �
Pi �iu and thus
kr�u uap�k�L���� � krXi
�i�u vi�k�L����� kX
i
r�i�u vi�k�L���� ! kXi
�ir�u vi�k�L�����
Now� since not more than M patches overlap in any given point x � �� the sumsPir�i�uvi� andPi �ir�uvi� also contain at mostM terms for any �xed x � �� Thus�
jPir�i�u vi�j� � MP
i jr�i�u vi�j� and jPi �ir�u vi�j� � MP
i j�ir�u vi�j�for any x � �� Hence� if we observe that supp�i � �i
kXi
r�i�u vi�k�L���� ! kXi
�ir�u vi�k�L���� �
MXi
kr�i�u vi�k�L���� ! MXi
k�ir�u vi�k�L���� �
MXi
kr�i�u vi�k�L���i���! M
Xi
k�ir�u vi�k�L���i����
MXi
�C�G
�diam�i������i� ! C�
�����i�
�
which �nishes the proof� �
Remark ���� The constant M controls the overlap of the patches� In particular� notmore than M patches overlap in any given point x � � of the domain� The patches haveto overlap because the functions �i are supposed to form a su�ciently regular �here�Lipschitz� partition of unity� Condition ��� expresses the fact that we need to controlthe gradient of the partition of unity functions �i if we are interested in H� estimates�Note that typically ���i� � C�diam�i����i� so that the terms in the sum of ���� are in asense balanced�The usual piecewise linear hat functions on a regular �triangular� mesh in two dimensionssatisfy the above conditions of a �M�C�� CG� partition of unity� actually�M � �� C� � ��and condition ��� is satis�ed because of the regularity of the mesh� i�e�� the minimumanglecondition satis�ed by the triangulation� Similarly� the classical bilinear �nite element
�
functions on quadrilateral meshes form a �M�C�� CG� partition of unity �M � �� C� ����The PUFEM has approximation properties very similar to the usual h and p version if thelocal approximation spaces Vi are chosen to be spaces of polynomials� In fact� if the localapproximation spaces consist of polynomials of �xed degree p and the approximation in Viis achieved through the smallness of the patch �i� the method behaves like the h version�If the patches are kept �xed and the local approximation is achieved by increasing thedegree p of the polynomials� which comprise the local spaces Vi� the method behaves likethe p version� In this sense� the PUFEM is a generalization of the h and p version�
� Examples of Local Approximation Spaces
Let us consider a few examples of systems of functions which have good approximationproperties for the solutions to a given di erential equation and additionally solve thedi erential equation themselves� A minimal condition on such a system is that it be densein the set of all solution� We will see that these systems are not unique and that there aremany dense system for a given di erential equation� The choice of a particular systemthus depends on practical aspects �cost of constructing the functions� ease of evaluationof the functions� i�e�� cost of construction of the sti ness matrix� conditioning numberof the resulting sti ness matrix� and theoretical aspects �optimality of the system� seeremark ��� below��
��� Laplace�s Equation
Let us begin with Laplace�s equation
"u � � ���
on a bounded Lipschitz domain � � R�� The classical approximation theory in L� withharmonic polynomials leads to results of the following form�
Theorem ��� �Szeg�o Let � � R� be a simply connected� bounded Lipschitz domain�
Let #� �� � and assume that u � L��#�� is harmonic on #�� Then there is a sequence�up��p�� of harmonic polynomials of degree p such that
ku upkL���� � Ce��pkukL������kr�u up�kL���� � Ce��pkukL�����
where �� C � � depend only on �� #��
Proof� See �� �� ���� �
Theorem ��� Let � be a bounded Lipschitz domain� star�shaped with respect to a ball�Let the exterior angle of � be bounded from below by ��� � � � Assume that u �Hk���� k � �� is harmonic� Then there is a sequence �up��p�� of harmonic polynomialsof degree p such that
ku upkHj��� � C�diam��k�j�ln p
p
���k�j�
kukHk���� j � �� � � � � �k�
where C � � depends only on the shape of � and k�
See �� for a proof of theorem ��� Note that typically � � � and that for domains withre�entrant corners� � can be signi�cantly less than ��Remark ���� The restriction in theorem �� that � be star�shaped with respect to aball is not a big constraint for our purposes because we are interested in local estimateson patches and the patches are typically chosen to be star�shaped�
Remark ���� We note that the error estimates of theorem ��� are �up to the constantsinvolved� similar to the estimates one obtains for the approximation with full spaces ofpolynomials in that the dependence on p is essentially the same� However� since thenumber of harmonic polynomials of degree p is p ! � and the number of polynomialsof degree p is p�p��
�� the approximation with harmonic polynomials is �asymptotically�
better in terms of error versus degrees of freedom�
Remark ��� We formulated theorem �� in an H� framework� Similar results in anL� setting can be found in ���� for example� Those estimates also exhibit the loss in therate of the approximability when the domain � has re�entrant corners�
Remark ���� Harmonic polynomials are not the only system of functions whichare dense in the class of solutions to Laplace�s equation� For example� the systemsfRe enz� Imenz jn � N�g� or fRe z�n� Im z�n jn � N�g �if � �� ��� or the system of rationalfunctions are dense in the set of solutions of Laplace�s equation� The system of harmonicpolynomials is optimal in the sense of n�width for the approximation of rotationallyinvariant spaces of harmonic functions on discs �see ������
��� Elasticity Equations
The solutions of the equations of linear elasticity �in the absence of body forces� in twodimensions can be expressed in terms of two holomorphic functions �see ������ Let usconsider the case of plain strain on a bounded Lipschitz domain � � R
� and let �� be the Lam$e constants of the material �for the case of plain stress� replace in whatfollows � by �� � ���� ! ��� The displacement �eld �u� v� can be expressed by twoholomorphic functions �� ��
�u�x� y� ! iv�x� y�� � ��z� z���z� ��z� ����
where � �� ! ����� ! � and we set z � x ! iy� For a given displacement state�the holomorphic functions �� � are unique up to the normalization of ��z�� � � in a
point z� � �� Thus� we may approximate the displacement �eld �u� v� by �generalizedharmonic polynomials�
�u! iv� � �p�z� z��p�z� �p�z� ����
where the functions �p� �p are complex polynomials of degree p
�p�z� �pX
n��
an�z z��n
�p�z� �pX
n��
bn�z z��n
with complex coe�cients an� bn� In a real formulation� the displacements u and v areobtained by taking the real and imaginary parts of the elements of the space V �as avector space over R of dimension ! �p�
V � span f�� i� �z z��n� i�z z��n�
�z z��n n�z z���z z��n���
i �z z��n in�z z���z z��n�� jn � �� � � � � pg�
The approximation properties of these �generalized harmonic polynomials� are very sim�ilar to the approximation properties of the harmonic polynomials for the approximationof solutions of Laplace�s equation� Obviously� in the case that the displacement �eld sat�is�es the elasticity equations on a domain #� �� �� the estimates of theorem ��� producesimilar estimates for the error in the displacement �eld and stress �eld for the approxima�tion with �generalized harmonic polynomials�� The analogous theorem to theorem ��takes the form
Theorem ��� Let � � R� be a bounded Lipschitz domain� star�shaped with respect to
a ball� Let the exterior angle of � be bounded from below by #��� Assume that thedisplacement �eld �u� v� � Hk���� k � �� Then �u� v� can be approximated by �generalizedharmonic polynomials of degree p such that
k�u! iv� � �p �z z����p �p�kHj��� � C�diam��k�j�ln p
p
����k�j�
k�u� v�kHk���
for j � �� �� The constant C depends only on the shape of � and k�
Proof� The proof can be found ��� A density assertion for these �generalized harmonicpolynomials� in the space of solutions of the elasticity equations can also be found in ����under stronger assumptions� however�� �
Remark ���� As in the example with Laplace�s equation� we are not restricted tousing �harmonic polynomials�� Analogous systems based on functions of the form enz�or polynomials in ��z are also dense in the set of solutions of the elasticity equations�
Remark �� � The theory can be extended to problems with certain loads� In manypractical applications the load is simple �constant� polynomial� and an explicit particular
�
solution of the elasticity equations is known� Thus� augmenting the space V by thisparticular solution allows us to deal with these problems successfully in the frameworkof approximating the sought solution by functions which solve the di erential equation�
��� Helmholtz�s Equation
In this section let us consider the Helmholtz equation on a bounded Lipschitz domain� � R��
"u! k�u � � on � � R� ����
where k � � is the wave number� For this problem we discuss two sets of functions whichhave good approximation properties for the general solution of ����� De�ne �generalizedharmonic polynomials� of degree p by
V V �p� � span fe�in�Jn�kr� jn � �� � � � � pg ����
where we used polar coordinates �r� �� and the functions Jn are the usual Bessel functionsof the �rst kind �see� e�g�� ����� The nomenclature �generalized harmonic polynomials�comes from the fact that these functions are the direct analogues of harmonic polynomialsvia the theory of Bergman and Vekua ����� ������ In fact� the approximation results for theapproximation of harmonic functions with harmonic polynomials carry over to the caseof the approximation of the solutions of ���� with �generalized harmonic polynomials��
Theorem �� Let � be a bounded Lipschitz domain� star�shaped with respect to a ball�Let the exterior angle of � be bounded from below by �� and assume that u � Hs����s � �� solves ���� Then there are functions up � V V �p� such that
ku upkHj��� � C��� s� k�
�ln p
p
���s�j�
kukHs��� j � �� �
where C��� s� k� � � depends only on �� k� and s�
Proof� see ��� �
As in the case of the approximation of solutions to Laplace�s equation� there are manyother alternatives to the choice of �generalized harmonic polynomials�� For example� onecan approximate the solutions of ���� with systems of plane waves�
W �p� � span fexp �ik�x cos �j ! y sin �j�� j �j � �
pj� j � �� � � � � p �g� �� �
One can show that these systems of plane wave have approximation properties which arevery similar to the approximation with �generalized harmonic polynomials��
Theorem ��� Under the same assumptions as in theorem �� there are functions up �W �p� such that
ku upkHj��� � C��� s� k�
�ln� p
p
���s�j�kukHs��� j � �� �
where C��� s� k� � � depends only on �� k� and s�
��
What are the di erences between the generalized harmonic polynomials and the systemsof plane waves% Just as the harmonic polynomials were optimal in the sense of n�widthfor the approximation of rotationally invariant spaces of harmonic functions on discs� thegeneralized harmonic polynomials are optimal in the sense of n�width for rotationallyinvariant spaces of solutions of ���� for the special case of � being a disc�An advantage of systems of plane waves is that they might be easier to use in practicalapplications� Plane waves can be written as products of functions of x and of y only�thus� if the patches consists of rectangles aligned with the coordinate axes� then theintegrals appearing in the sti ness matrix can written as products of one dimensionalintegrals and evaluated cheaply� This observation has been exploited in section ���� Letus �nish this section by mentioning that these �generalized harmonic polynomials� andthe systems of plane waves lead to exponential rates of convergence if the function u isanalytic up to boundary�
Theorem ��� Let � � R� be a simply connected� bounded Lipschitz domain� Let #� ��� and assume that u � L��#�� solves the homogeneous Helmholtz equation on #�� Then
infup�V V �p�
ku upkH���� � Ce��pkukL�����inf
wp�W �p�ku wpkH���� � #Ce���p� lnpkukL�����
where C� #C� �� and #� depend only on �� #�� and k�
��� Change of Variables Techniques� Rough Coe�cients and
Elasticity Equations with Corners
The idea of the PUFEM is to enable the user to employ functions with good localapproximation properties� These functions do not necessarily have to solve the di erentialequation� In fact� it can sometimes be too costly to create �optimal� functions� Onemethod to create functions which have good local approximation properties is obtainedby an appropriate change of variables� Let us assume that the change of variables x �� #xtransforms the sought solution u into a function #u which is smoother than u� Then� thistransformed function #u can be approximated well by polynomials #P �#x�� This suggeststhat a good choice for the approximation of u are the mapped �polynomials� P �x� � #P �#x�where the functions #P are polynomials�This idea has been analyzed for a model problem with unilaterally rough coe�cients in���� �the next section considers in detail the one dimensional analogue of the problemconsidered in ������The idea of exploiting the improved approximation properties of mapped �polynomials�has been applied very successfully to the problem of the elasticity equations with singu�larities ������ ����� The natural change of variables �in a two dimensional setting� is aconformal map which makes corner singularities or singularities arising at interfaces lesspronounced� The mapped function can be approximated well by polynomials� Mappingthe polynomials back under this conformal map leads to the ansatz functions used�
��
�� The Choice of the Partition of Unity Functions
In the preceding subsections� we described various choices of local approximation spaceswhich have better approximation properties than the spaces of polynomials of degree p�Let us now turn to the problem of the choice of the partition of unity which puts a givenset of local approximation spaces together to produce a conforming global space� Theconditions on the partition of unity are very weak� a Lipschitz partition of unity su�cesto construct a subspace of H� according to theorem ���Let us consider a domain � � R�� One possible choice of a partition of unity is a collectionof �nite element functions� Let #� � � be any domain on which a mesh �consisting oftriangles or rectangles� say� has been de�ned� The usual piecewise linear or bilinearhat functions associated with the nodes of this mesh form a partition of unity for #�and therefore for � as well� The supports of these hat functions can then be taken asthe patches �i� If the mesh satis�es a minimum angle condition� this partition of unitysatis�es all the requirements of theorem ��� This particular choice has been made forthe numerical example of section ����A more general choice of a partition of unity is given by the following procedure� Letf�ig be a collection of overlapping patches which cover � and let f�ig be a collection offunctions which are supported by the patches �i� Then the normalization
�i ��iPj �j
���
yields a partition of unity subordinate to the cover f�ig� Note that for given i the sumin ��� actually only extends over those j which satisfy �i � �j �� �� The functions �i
inherit the smoothness of the functions �i and thus this normalization technique givesone possible construction of �nite element spaces with higher regularity� for example�subspaces of H��We have seen in the introduction that a �nite element method is completely determinedby the bilinear form and the �nite dimensional trial and test spaces� In order to solve ��in practice� we have to �nd bases for the test and trial spaces� Since the �nite elementspaces V constructed by the PUFEM are of the form V �
P�iVi where the �i are a
partition of unity and the Vi are the local approximation spaces� it is natural to seek abasis of V based on bases of the spaces Vi� If fvi�p j p � �� � � �g are basis functions of thelocal spaces Vi� one can hope that the functions
B � f�ivi�pg ����
form a basis of V � However� there are a few cases� where the set B is not linearlyindependent� In order to see this� let us consider a one dimensional example� De�ne� � ��� ��� h � ��n� xi � ih� i � �� � � � � n� �i � �xi h� xi ! h�� and let �i be theusual piecewise linear hat function associated with the node xi� Now choose for the localapproximation spaces Vi � span f�� x� � � � � xpg� p � N� The PUFEM space V is thenprecisely the space of continuous functions which are piecewise polynomials of degreep! �� i�e�� dimV � n�p! �� ! �� On the other hand� the set B contains n�p! �� ! p! �elements� Thus� B cannot form a basis of V � Of course� this particular example issomewhat contrived and in general the set B will form a basis of V � However� this
�
example shows that we may have to expect that the elements of B could be nearlylinearly dependent which would lead to badly conditioned sti ness matrices�One way to ensure that the sets B of the form ���� are linearly independent is to constrainthe partition of unity in such a way that each function �i is identically � on a subset of�i and all other functions �j vanish on this subset�The linear dependencies in the one dimensional example above can be removed by aslight change of the partition of unity functions� It is enough to change those partition ofunity functions which are close to boundary� Since we will use this particular partitionof unity for the numerical example in section ���� we describe it in more detail�
xi � ih i � �� � � � � n ��� � ��� h��i � �xi h� xi ! h� i � � � � � � n
�n�� � �� h� ��
�� �
�� if x � ��� h�� x�h
hif x � �h� h�
�i �
�� ! x�xi
hif x � �xi h� xi�
� x�xih
if x � �xi� xi ! h�i � � � � � � n
�n�� �
�� ! x����h�
hif x � �� h� � h�
� if x � �� h� ��
���
� A Robust Method for an Equation with Rough
Coe�cients
��� Construction of Robust Local Approximation Spaces
In this section� we want to construct a robust method for the approximation of thesolution of an equation with rough coe�cients� As a model problem let us consider theelliptic boundary value problem
Lu � �a�x�u��� ! b�x�u � f on � � ��� ��u��� � u��� � �
���
where the coe�cients a� b � L���� satisfy
� a� � a�x� � kakL� � � � b�x� � kbkL� on ��
We assume that the function f � L�� Observe that the solution u of ��� and thefunction au� are Lipschitz continuous� i�e��
u� au� � W �������
However� if a is merely in L�� we cannot expect the solution u to be in some H������� � �� Thus� the classical piecewise polynomial �nite element spaces may perform verypoorly� In fact� the following result holds�
��
Lemma �� Let b � �� f � � in problem ��� and let &�n� be any sequence of numberswhich decreases monotonically to �� Then one can �nd a function a � L� with � �a�x� � and a constant C � � such that for any n dimensional space Vn of continuous�piecewise linear functions
infun�Vn
ku unkH���� � C&�n�� ��
Proof� �� �
The lemma shows that the usual FEM may converge arbitrarily slowly �as the number ofdegrees of freedom n is increased� if the coe�cient a is su�ciently rough� Note that ��holds for all spaces of continuous� piecewise linear functions� and thus we cannot improvethe rate of convergence by choosing the meshes judiciously� In practice this means thatthe classical FEM breaks down for these rough coe�cients because �convergence� is onlyachieved for extremely small mesh sizes h�Remark ��� The case that the coe�cients a� b are smooth but highly oscillatory �i�e��large derivatives� is also covered by the ensuing theory� When the coe�cients are smoothbut highly oscillatory� the exact solution u may be smooth �in H�� say�� but kukH���� isso large that the asymptotic behavior of the FEM is visible for very small mesh sizes only�The special ansatz functions constructed below circumvent this phenomenon and lead torobust �nite element methods which behave like the usual FEM for smooth coe�cientsa� b �with reasonable bounds on the derivatives��
The goal of this subsection is to construct �local� approximation spaces for the approx�imation of u which are robust� We construct spaces with any desired order of approx�imability �for su�ciently smooth right hand side f � the coe�cients a� b� however� arestill assumed to be merely in L��� In proposition �� we exhibit such spaces� However�since the functions of proposition �� are the solutions of auxiliary problems� which arenot necessarily easier to solve than the original problem� we present approximations ofthese functions in theorem ��� which have approximation properties as good as those ofproposition ���
De�ne
B �kbkL�a�
and let us consider the approximation of u on an interval I � � of length h by twofunctions u�� u� which form a fundamental system for L� i�e�� any solution v of theequation Lv � � can be expressed as a linear combination of u�� u��
Proposition �� �Approximation with fundamental systems Let u be the solu�tion of ���� I � � be an interval of length h� and let u�� u� be a fundamental system forL� Under the assumption that Bh � � �� there is uh � V � span fu�� u�g such that
ku uhkL��I� � �
a��� ��h�kfkL��I�
k�u uh��kL��I� � �
a��� ��hkfkL��I��
��
Proof� Fix x� � I� Choose uh � V such that the function e � u uh satis�es
Le � f e�x�� � � �ae���x�� � ��
Then we have an explicit formula for the error e
�ae���x� � Z x
x�f be dt� ���
Since e�x�� � �� we have kekL��I� � hke�kL��I� and hence ��� allows us to bound
We note that the estimates of proposition ��� are robust in the following sense� Theexact solution� in spite of being merely in W ���� can be approximated with accuracyO�h� independently of the roughness of the coe�cients a and b� Only the bounds a� andkbkL� enter in the estimates�Proposition ��� gives local approximation spaces which are �rst order accurate� Let usnow construct local approximation spaces which have higher order of accuracy �assumingthat the right hand side f is su�ciently smooth�� To that end we will augment the spaceV of proposition ��� by particular solutions to certain right hand sides�
Proposition �� �Approximation with augmented fundamental systems Let ube the solution of ���� I � � be an interval of length h� x� � I be a reference point�and let u�� u� be a fundamental system for L� Let vi� i � N�� be functions such thatLvi � �x x��i� For p � N� � f�g de�ne the space
Vp ��span fu�� u�� v�� � � � � vpg if p � N�
span fu�� u�g if p � ��
Under the assumption that Bh � � � and f � Cp����� there is uh � Vp such that
ku uhkL��I� � �
a��� ���p! ��'hpkf �p��kL��I�
k�u uh��kL��I� � �
a��� ���p! ��'hp�kf �p��kL��I��
��
Proof� The case p � � has been handled in proposition ���� Let therefore p � N��Taylor�s theorem allows us to write f �
Ppn�� fn�x x��
n ! R�x� where kRkL��I� �hp��
�p���kf �p��kL��I�� Then the function e � u Pp
n�� fnvn satis�es Le � R on I� Usingproposition ��� we can approximate e with the functions u�� u� and arrive at the desiredestimates� �
Proposition �� permits us to construct robust methods of any desired order �assumingthat the right hand side f is su�ciently smooth� if we can �nd the local approximationfunctions u�� u�� v�� � � �� In the special case b �� these functions are explicitly available�
u� � � u��x� �Z x
x�
�
a�t�dt vi�x� � �
i! �
Z x
x�
�t x��i�
a�t�dt�
In the general case� b � �� �nding u�� u�� and the vi amounts to solving appropriateauxiliary problems on I� In practice� we have to �nd approximations to the functionsu�� u�� vi� In the rest of this section� we will describe one method to approximate thesefunctions and analyze how accurate these approximations have to be� For the approxi�mation of these functions� we will use the fact that they can be written as the solutionsof appropriate Volterra integral equations which can be solved by an iterative method�We will see that only few iterations are necessary to yield satisfactory approximations ofthe functions u�� u�� vi�For the remainder of the section� let I � � be an interval of length h and let x� � I bea reference point in I� Let us consider the initial value problem
Lw � g � L��I� w�x�� � w� �aw���x�� � w�� ���
The function w is the solution of the following Volterra integral equation
w � Kw ! #w ���
where the operator K and the function #w are de�ned by
�Kw��x� �Z x
x�
�
a�t�
Z t
x�b�� �w�� �d�dt ���
#w�x� � w� ! w�
Z x
x�
�
a�t�dt
Z x
x�
�
a�t�
Z t
x�g�� �d�dt� � �
The theory of Volterra integral equations �see� e�g�� � �� allows us to expand the solutionoperator �I K��� in a Neumann series� and we can write
w ��Xn��
Kn #w�
We introduce now approximations to the exact solution w by partial sums of this series�
wN ��PN
n��Kn #w if N � N�
� if N � ����
We need to estimate wwN � The next two lemmas clarify the approximation propertiesof the approximants wN �
��
Lemma �� Let the operator K be de�ned as in ���� Then for any w � L��I� and anyn � N� we have
j�Knw��x�j � Bnjx x�j�n�n�'
kwkL��I�
j�Kn�w���x�j � Bn�jx x�j�n��n ! ��'
kwkL��I�
where again B � a��� kbkL��Proof� The �rst estimate is the classical estimate for Volterra integral equations �in aC� setting� and may be proved by induction� The second estimate follows from the �rstone with the observation
j�KKnw���x�j ������ �
a�x�
Z x
x�b�t��Knw��t�dt
����� � B
����Z x
x�j�Knw��t�j dt
���� ��
Remark ��� Lemma �� shows that the �xed point equation ��� can be solved by aNeumann series expansion in an C� or an W ��� setting� The Neumann series convergesfor any h � ��
Lemma �� Let w be the solution of the �xed point problem ��� and let wN be theapproximation given by ��� for N � N� � f�g� Then
kw wNkL��I� � h�N�C��N�h�B�k #wkL��I�
k�w wN ��kL��I� ��h�N�C��N�h�B�k #wkL��I� if N � N�
k #w�kL��I� ! hC���� h�B�k #wkL��I� if N � �
where C�� C� are de�ned by
C��N�h�B� �BN�
�N ! �'
�Xn��
�N ! �'
�N ! n ! �'�Bh��n
C��N�h�B� �BN�
�N ! ��'
�Xn��
�N ! ��'
�N ! n ! ��'�Bh��n�
Proof� We can write w wN �P�n�N�K
n #w and use the bounds on the operators Kn
obtained in lemma ��� �
Remark �� Under the assumption #w � W ����I�� #w�x�� � �� the estimate on�w wN �� can be formulated in the following� more compact form�
k�w wN ��kL��I� � h�N�C�N�h�B�k #w�kL��I�
where C is given by
C�N�h�B� ��C��N�h�B� if N � N�
� ! h�C���� h�B� if N � ��
�
Remark ��� Under the assumptions Bh � � �� h �� we can easily bound C�� C�
by
C��N�h�B� � BN�
�N ! �'
�
� �hfor N � N� � f�g
C��N�h�B� � BN�
�N ! ��'
�
� �hfor N � N��
This analysis of the �xed point problem ��� is now the tool for the approximation of afundamental system u�� u� and for the approximation of particular solutions vi� Let u��u�� vi be given by
which are solutions of problem ��� for appropriately chosen w�� w�� and g� Let uN� � uN� �
and vNi � N � N� �f�g� be the approximations to the exact solutions as de�ned by ���Then� the following lemma holds�
Lemma �
ku� uN� kL��I� � h�N�C��N�h�B�
k�u� uN� ��kL��I� � h�N�C��N�h�B� for N � N�
ku� uN� kL��I� � h�NC��N�h�B��
a�
k�u� uN� ��kL��I� � h�N�C�N�h�B�
�
a�
kvi vNi kL��I� � h�N�iC��N�h�B��
a��i! ���i! �
k�vi vNi ��kL��I� � h�NiC�N�h�B��
a��i! ���
Proof� The proof follows directly from lemma ��� and remark ���� �
We would like to construct an approximation of the space Vp of proposition ��� Lemma ���enables us now to calculate how many terms of the Neumann expansion su�ce� Recallthat the error estimate of proposition �� for the approximation in Vp is O�hp�� �for theerror in the derivative�� The approximations uN� � v
N� � and vNi have to be calculated with
the same accuracy� This gives for the number of terms�
N� � p! �
for the approximation of u�
N� � p
for the approximation of u�
#Ni � p i �
for the approximation of vi
�
where N�� N�� #Ni � N� � f�g� Choosing the smallest N�� N�� and #Ni such that thesethree inequalities are satis�ed� we can de�ne
We now show that the space #Vp has indeed the desired approximation properties� i�e��the approximation properties of #Vp are essentially the same as those of the spaces Vp�
Theorem �� �approximate augmented fundamental system Let I � � be aninterval of length h� x� � I be any reference point in I� and let u be the solution of ����Let p � N� � f�g� f � Cp����� #Vp be de�ned as in ��� and assume that Bh � � ��Then there is uh � #Vp such that
ku uhkL��I� � hpC�p�B� a�� ��kfkCp�����
k�u uh��kL��I� � hp�C�p�B� a�� ��kfkCp�����
where C�p�B� a�� �� depends only on p� B� a�� �� and ��
Proof� The proof follows very closely the proof of proposition ��� Let us write
f�x� �pX
n��
f �n��x��
n'�x x��
n !R�x�
where the remainder R�x� satis�es kRkL��I� � hp��
�p���kf �p��kL��I�� If we agree to assignthe empty sum the value �� the estimate for R also holds for p � �� The approximantof proposition �� could be chosen to be �cf� also remark ���
uap � u�x��u� ! �au���x��u� !pX
n��
f �n��x��
n'vn�
Because the functions vn satisfy vn�x�� � �av�n� �x�� � �� the error r � u uap satis�es
Lr � R r�x�� � � �ar���x�� � ��
Let us approximate u in #Vp by
uh � u�x��uN�
� ! �au���x��uN�
� !pX
n��
f �n��x��
n'v�Ni
i �
��
and we get the following representation for the error�
u uh � u�x���u� uN�
� � ! �au���x���u� uN�
� � !pX
n��
f �n��x��
n'�vi v
�Ni
i � ! r�
From lemma ��� with N � �� we can bound r
krkL��I� � C���� h�B�h�
a�kRkL��I� � C���� h�B�
hp
a��p! ��'kf �p��kL��I�
kr�kL��I� � C��� h�B�h
a�kRkL��I� � C��� h�B�
hp�
a��p ! ��'kf �p��kL��I��
Applying the estimates of lemma ��� to the remaining terms of the error represen�tation �nishes the proof� if we observe that kukL��I�� kau�kL��I� can be bounded byC�a�� B���kfkL����� C�a�� B��� depends only on a�� B� and � according to standardregularity theory� �
Remark ��� The approximation properties of the space #V�� can be understood withthe ideas of section ��� as well� If one introduces the change of variables #x �
R x�
�a�t�dt�
then problem ��� is transformed to a problem of the form
#u�� !#b#u � #f
where #b� #f are still in L� and hence #u � W ���� The elements of #V�� transform to linearfunctions� Therefore� the approximation of u in #V�� can be expected to behave like theapproximation of a W ��� function by linear functions�
��� Construction of the Global Finite Element Space
We will now construct a global conforming �nite element space from the spaces #Vp �cf������ which have good local approximation properties for the approximation of thesolution of ���� We proceed as outlined in section � Let ��i�Ni�� be a covering of� � ��� �� satisfying the overlap condition� Let ��i�Ni�� be a �M�C�� CG� partition ofunity associated with this covering ��i�� The local approximation spaces Vi are given bytheorem ��� as follows� In each patch� we choose a reference point zi � �i �which playsthe role of the point x� of theorem ����� For p � N� � f�g� the local approximationspaces Vi � Vi�p� are then taken as the spaces #Vp of ��� with reference point zi insteadof x�� Theorem ��� immediately gives for the local approximation properties �expressedin the notation of theorem ���
���i� � C�p�B� a�� �����diam�i�p���kfkCp�����
���i� � C�p�B� a�� �����diam�i�p����kfkCp������
We de�ne the global approximation space V � V �p� �PN
i�� �iVi�p�� Hence� for u solving���� there is uh � V �p� such that
ku uhkL�����pMC�C�p�B� a�� ����kfkCp�����
�NXi��
�diam�i���p��
����
�
k�u uh��kL���� �
qM�C�
G ! C���C�p�B� a�� ����kfkCp�����
�NXi��
�diam�i���p���
����
So far we have not dealt with the essential boundary conditions at x � � and x � ��However� they are easily enforced by a judicious choice of the reference point for thepatches �i close to the boundary� i�e�� �i � �� �� �� For these patches� we choose thereference point zi to be the boundary point and then simply leave out the approximationsuN�
� to u� because all the other elements of #Vp vanish at the reference point� The �niteelement space V �p� is thus a subspace of H���� and satis�es the boundary conditions�i�e�� it is a conforming �nite element space�Let us give a more concrete example of the abstract procedure given above for the con�struction of the global space V �p�� Let n � N� h � ��n and de�ne the patches �i andthe partition of unity �i as in equations ���� The local approximation spaces Vi�p�associated with the patches �i are given by ��� where the reference point in each patch�i is chosen to be the node xi for i � � � � � � n� For i � � the reference point is chosento the left boundary point x � � and for i � n � the reference point is chosen to beright boundary point x � �� The approximation space V��p� and Vn���p� associated withthe �rst and last patch are constrained to satisfy the essential boundary conditions byomitting the approximations to u�� For example� the two simplest spaces are
V ��� � span f���x�Z x
�
�
a�t�dt� �n���x�
Z x
�
�
a�t�dt�
�i�x�� �i�x�Z x
xi
�
a�t�dt j i � � � � � � n g ����
V ��� � span f���x�Z x
�
�
a�t�dt� ���x�
Z x
�
t
a�t�dt�
�n���x�Z x
�
�
a�t�dt� �n���x�
Z x
�
t �
a�t�dt�
�i�x�
�� !
Z x
xi
�
a�t�
Z t
xib�� �d�dt
��
�i�x�Z x
xi
�
a�t�dt� �i�x�
Z x
xi
t xia�t�
dt j i � � � � � � n g� ����
And the above theory gives that the spaces V ���� V ��� approximate the solution u of��� such that
infuh�V ����
ku uhkL���� ! hk�u uh��kL���� � C�B� a�� ��kfkL����h
� ���
infuh�V ���
ku uhkL���� ! hk�u uh��kL���� � C�B� a�� ��kfkC����h
����
where the constant C�B� a�� �� depends only on B� a�� and � if Bh � � �� Let us notethat
dimV ��� � �n �� ! dimV ��� � ��n �� ! �� ����
�
Figure �� Approximation in V ��� �!�� V ��� �o�� and Vpoly���
100
101
102
103
104
10−12
10−10
10−8
10−6
10−4
10−2
100
degrees of freedom
rel.
erro
r in
ene
rgy
N=4096; b=0; a continuous
��� Numerical Example
In this subsection� we apply the above constructed �nite element spaces to a concretedi erential equation� We consider
Lu � �a�Nx�u��� ! bu � f�x� on � � ��� ��u��� � u��� � �
����
where the function a is ��periodic� N � N large� and the coe�cient b is either b � � orb � �� The right hand side f is taken to be f � x for b � � and f � � for b � �� For the��periodic function a� we consider two cases�
a��x� ��
! cos��x�
a��x� �
�� if x � ��� ��� if x � ���� ���
The solution of ���� is in H���� �even piecewise C�� for both choices of the coe�cient a�However� the solution is rough in our terminology as is has very large higher derivatives�Associated with this problem is the notion of an �energy�
kuk�E �Z �
�a�Nx�ju�j� ! bjuj�dx
and an �energy� norm� which is the square root of the energy�
Figure � Approximation in V ��� �!�� V ��� �o�� and Vpoly���
100
101
102
103
104
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
degrees of freedom
rel.
erro
r in
ene
rgy
N=524288; b=1; a continuous
The typical behavior of the classical piecewise polynomial �nite element methods for thisparticular problem is to converge �in the energy norm� for very small mesh size only�namely when the mesh size h is so small that the �nite element space can resolve theoscillation of the coe�cient a� The classical �nite element methods therefore convergefor h N�� only�By the method outlined in the preceding subsection� we can create robust approximationspaces of any desired order for the approximation of ����� However� we restrict ourselveshere to the two spaces V ���� V ��� de�ned in ����� ����� For comparison� let us introducea third type of spaces� namely� a space where the local approximation spaces consist ofpolynomials� Using the same partition of unity f�ig as in the construction of V ����V ��� �cf� ����� we de�ne
This space Vpoly contains all piecewise linear functions and is a subset of the usual piece�wise quadratic �nite element space� It will therefore serve as a comparison of the usual�nite element method with our robust spaces�Fig� � and show the performance of the three spaces V ���� V ���� and Vpoly for thecoe�cient a� for the cases b � �� N � ����� and b � �� N � �� whereas �g� � and �correspond to the coe�cient a� for the cases b � �� N � ����� and b � �� N � ���In all the graphs� the mesh size ranges from h � �
� to h � ��� �� ���� relates these mesh
sizes to the number of degrees of freedom� in particular� the number of degrees of freedomis proportional to ��h for both V ��� and V ���� Therefore� estimates ���� ���� yield
�
Figure �� Approximation in V ��� �!�� V ��� �o�� and Vpoly �(�
100
101
102
103
104
10−12
10−10
10−8
10−6
10−4
10−2
100
degrees of freedom
rel.
erro
r in
ene
rgy
N=4096; b=0; a jumps
bounds of the formrel� error in energy � Cdof��� Cdof�� �� �
for the approximation in V ��� and V ���� respectively� The size of the constant C isindependent of the roughness of the coe�cient a� i�e�� it is independent of the numberN � We can see in �g� ��� that these rates of convergence are actually attained and thatthe method is robust� Estimates �� � hold for very few degrees of freedom and the goodbehavior of the method is independent of N �the PUFEM performs equally well for thecases N � ���� and N � ���� The spaces Vpoly behave in a totally di erent way�Since the graphs only cover the range h � �
�to h � �
�� �� we still have h � N�� and
cannot expect the usual FEM to work� Indeed� the error stays almost constant over thewhole range�We considered two cases b � � and b � �� The di erence between those two cases liesin the fact that for b � � the spaces V ��� and V ��� are based on local approximationspaces which contain an exact fundamental system whereas in the case b � � the localapproximation spaces contain only an approximate fundamental system� We see� how�ever� that the approximate fundamental system is accurate enough not to upset the rateof convergence� just as the theory of section ��� predicts�Finally� let us mention that we chose a problem with periodic coe�cients for computa�tional convenience� In this particular case� the periodicity could be exploited in such away that the construction of the sti ness matrix and the evaluation of the right handside is achieved with an amount of work independent of the number N � the work is � upto a constant � the same as for the usual �nite element method for N � ��
�
Figure �� Approximation in V ��� �!�� V ��� �o�� and Vpoly �(�
100
101
102
103
104
10−12
10−10
10−8
10−6
10−4
10−2
100
degrees of freedom
rel.
erro
r in
ene
rgy
N=524288; b=1; a jumps
Table �� DOF necessary to obtain accuracy � in L� norm� k � ���� best p�w� linear QSFEM GLSFEM FEM
This numerical example shows that the PUFEM based on the local approximation spacesconstructed in section ��� leads to a robust method� The performance of the �nite elementspaces V ���� V ��� is independent of the roughness of the coe�cients of the di erentialoperator and their performance is comparable to the classical piecewise linear or quadratic�nite element spaces for a problem with smooth coe�cients�
� Helmholtz�s Equation and Concluding Remarks
�� Helmholtz�s Equation
In this section� we present an application of the PUFEM to the Helmholtz equation intwo dimensions� We consider the problem
"u k�u � � on � � ��� ��� ��� �� � R�
�nu! iku � g on �����
�
Table � DOF necessary to achieve various accuracies in L� for PUFEM with n � � andvarious other methods� k � ���
Table �� operation count for band elimination for PUFEM� k � �� error in H�� n � �p H� error NOP PUFEM� ��) ���D!� �� ) ��D!�� ���) ��D!��� ������) ���D!�
�
Figure �� The p version of the PUFEM
0 50 100 150 200 250 300 35010
-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
DOF
rel.
erro
r in
H^1
sem
i nor
m
rel. error in H^1 semi norm of PUFEM for k=32.0
n=1
n=2
where g is chosen such that the exact solution is a plane wave of the form
u�x� y� � exp fik�x cos � ! y sin ��g� � ��
���
In section ��� we discussed two types of local approximation spaces in for the approxima�tion of solutions of Helmholtz�s equation� We could take either the �generalized harmonicpolynomials� of ���� or the systems of plane waves �� �� In the numerical examples pre�sented here� we concentrate on the systems of plane waves �for a comparison of these twodi erent local spaces� see ���� The partition of unity for this particular problem is givenby piecewise bilinear hat functions� For n � N� the square � is subdivided into n � nsquares of side length h � �
n� With each of the �n ! ��� nodes �xi� yi� we associate a
piecewise bilinear hat function �i which vanishes in all nodes except �xi� yi�� The patches�i are taken to be the supports of these �i� The PUFEM is based on this partition ofunity and the local approximation spaces Vi are chosen to be the spaces W �p� of �� ��Remark ���� In this particular implementation we only used the space W �p� with pof the form p � �m! � m � N� to ensure that the exact solution of problem ��� is notin the PUFEM space�
In this application of the PUFEM� we have thus two parameters which in�uence theapproximation properties of the global �nite element space� namely� the mesh size of thepartition of unity� which is determined by n� and the size of the local approximation spacesVi� which is controlled by p� If the parameter p is �xed and the mesh size is variable� wetalk about the h version of the PUFEM� if the mesh is �xed and the approximation is
achieved by increasing the size of the local spaces �i�e�� by increasing p�� we talk aboutthe p version of the PUFEM� If both h and p are varied� we would then talk about thehp version of the PUFEM� The estimates on local approximability of theorem ��� let usexpect exponential rates of convergence as a p version� This exponential convergence ofthe p version of the PUFEM can be observed in �g� � for the cases n � � and n � �We will discuss the numerical results only brie�y� a more detailed analysis can be foundin ����� In tables ��� the PUFEM is compared with the usual Galerkin �nite elementmethod �FEM�� the generalized least squares �nite element method �GLSFEM� of ����and the quasi�stabilized �nite element method �QSFEM� of ����� Since all three methodsare based on piecewise linear functions on uniform grids� tables � and include thepiecewise linear best approximant for reference� In tables ���� we use the norm L� as theerror measure and analyze the case k � ���� Tables ��� deal with the case k � � andthe H� semi norm as the error measure� Tables � and show that the p version of thePUFEM needs markedly fewer degrees of freedom to achieve the same accuracy in L� asthe other methods� which are based on piecewise linear ansatz functions� This reductionin degrees of freedom translates in a reduction of the number of operations when thelinear system is solved using Gaussian elimination� This is demonstrated in table �� Intable � we list the various combinations of p and n which lead to the same accuracyof ��) in L�� Since we expect the PUFEM to exhibit exponential rates of convergenceas a p version but only algebraic rates as an h version� the number of operations issmallest for the largest mesh size h� In tables � and � we compare the operation countof the Gaussian elimination for the PUFEM with the operation count of the Galerkinmethod and the QSFEM� The linear systems in these latter two methods are solved bythe iterative method proposed in ���� We see that here again� the PUFEM performsbetter than the other two methods�We have seen that the PUFEM is superior to the other methods both in terms of errorversus degrees of freedom and error versus number of operations� Let us point out that thediscrepancy between the PUFEM and the other methods becomes larger as the accuracyrequirement is increased�Remark ���� We used systems of plane wave as local approximation spaces becausetheir speci�c structure and the particular form of the partition of unity allowed us tocreate the sti ness matrix cheaply� Therefore� the overall amount of work for the PUFEMis dominated by the operation count of the Gaussian elimination�
�� Concluding Remarks and Open Questions
We presented a new method which allows the user to include a priori knowledge about theproblem under consideration in the �nite element space� We illustrated this procedurein detail for a one dimensional model problem with rough coe�cients� For this onedimensional example� we constructed local approximation spaces which re�ect the roughbehavior of the solution� and the PUFEM enabled us to build a robust �nite elementmethod from these local spaces� A numerical example illustrated the robustness of themethod and thereby showed the superiority of the PUFEM over the classical FEM forthis particular kind of problem� With an application of the PUFEM to the Helmholtz
equation in two dimensions we demonstrated that the PUFEM can cope with highlyoscillatory problems in a very satisfactory fashion�We mentioned only very brie�y the other features of the PUFEM� Among them are theability to construct smoother space which are necessary for �nite element methods forhigher order di erential equations� Since the regularity of the PUFEM space is governedby the smoothness of the partition of unity� such smoother spaces are easily constructedwith the PUFEM� The �meshless� aspect of the PUFEM has also not been addressed inthis paper� This is a feature of the PUFEM which can be important for problems whichinvolve frequent remeshing such as the optimal placement of a fastener alluded to in theintroduction�We have seen that the PUFEM o ers a new� very promising approach to dealing suc�cessfully with non�standard problems where the usual �nite element methods fail or aretoo costly� Since the PUFEM is in still its infancy� there are also many open questionsabout implementational aspects which need to be addressed� Among them are�
�� The choice of a basis of the PUFEM space� We discussed this topic brie�y insection ���� It is an important issue because the condition number of the sti nessmatrix depends on the choice of the basis�
� The implementation of essential boundary conditions� We did not discuss this ques�tion because we concentrated on a one dimensional model problem where essentialboundary conditions can be enforced very easily�
�� The integration of the elements of the sti ness matrix� This is a di�culty whichthe PUFEM shares with all meshless methods� For the construction of the sti nessmatrix� one has to integrate shape functions against each other� Thus� the integra�tor has to be able to integrate e�ciently over the intersection of the supports ofthe shape functions� Since the shape functions are not necessarily tied to a mesh�the description of these intersections is potentially harder than in the usual FEM�However� speci�c choices of the partition of unity and�or appropriately designedintegrators should be able to cope with the integration issues successfully�
References
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�� I� Babu�ska and J� E� Osborn� private communication�
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�
��� I�S� Gradshtein� Table of Integrals� Series and Products� Academic Press� New York�����
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