AN ANALYSIS OF NEW FINITE ELEMENT SPACES FOR MAXWELL S EQUATIONS

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1 Journal of Matematical Sciences: Advances and Applications Volume 5 8 Pages -9 Available at ttp://scientificadvances.co.in DOI: ttp://d.doi.org/.864/jmsaa_7975 AN ANALYSIS OF NEW FINITE ELEMENT SPACES FOR MAXWELL S EQUATIONS Department of Matematics Hannam Universit Ojeong-dong Daedeo-gu Daejeon 6-79 Republic of orea lm97j@aist.ac.r Abstract In tis paper we introduce four different finite element spaces on eaedron for Mawell s equations. We also describe interpolation operators tat map from suitable subspaces H ( Ω ) H( curl Ω) H( div Ω) and te space L ( Ω) into te appropriate finite element spaces. For te error analsis we prove tat te spaces and interpolation operators are lined b te discrete de Ram diagram.. Introduction In performing te analsis of finite element metods for Mawell s equations it is necessar to present four different finite element spaces. Te most obvious requirement is te curl conforming edge elements suitable for discretiing te basic energ space for electromagnetics. Te most common eample is Nedelec elements given in [7]. Recentl we Matematics Subject Classification: 65N 65N. ewords and prases: H conforming element edge element divergence conforming element L conforming element de Ram diagram. Received June 8 8 Scientific Advances Publisers

2 ave introduced a new famil of curl conforming finite elements on parallelepiped grids in [5] wit fewer degrees of freedom tan te eisting ones. To anale edge elements we also need to present te divergence conforming elements and scalar finite elements. Well-nown eamples for te divergence conforming element spaces are te Raviart- Tomas-Nedelec (RTN) space and te Brei-Douglas-Duran-Fortin (BDDF) space [4]. We ave introduced a new famil of divergence conforming elements ling between RTN and BDDF spaces [5]. Altoug BDDF space as less degrees of freedom tan our new element space tere is no finite element corresponding to te curl conforming finite element space. For te scalar finite elements we also ave presented a new elements in [6]. For completeness we sall present a related new finite element famil in L ( Ω ). We will describe interpolation operators π r ω and P tat map from suitable subspaces U H ( Ω) V H( curl Ω) W H( div Ω) and te space L ( Ω) into te appropriate finite element spaces and prove te interrelationsip of te spaces lined b te discrete de Ram diagram.. Four Finite Element Spaces In tis section we will describe four different finite element spaces on eaedra wit edges parallel to te coordinate aes. First we generate a regular finite element mes τ { } > covering te domain Ω suc tat () Ω were Ω denotes te closure of Ω ; τ () for eac τ is an open set wit positive volume; () if and are distinct elements in τ ten ; (4) eac τ is a Lipscit domain.

3 AN ANALYSIS OF NEW FINITE ELEMENT SPACES For eac element we define te parameters and ρ suc tat diameter of te smallest spere containing ρ diameter of te largest spere contained in ten τ ma so tat te inde denote te maimum diameter of te elements τ. An τ can be obtained b mapping te reference element tpicall unit cube using an affine map F : suc tat F ( ) and F ( ) B b were B is an invertible diagonal matri and b is a vector. We sall be concerned wit mapping between functions defined on te reference element and te target element. For a simple scalar function p defined on we obtain a corresponding function p on b p F p were denotes composition of functions. And a simple calculation using te cain rule sows tat T ( p) F B p were is te gradient wit respect to. However vector functions must be transformed in a more careful wa to conserve teir properties. For v H ( curl ) we define v on b v T F B v. And if w H ( div ) ten we define w b w F B. det( B ) w

4 4 Because of te simplicit of te mapping altoug we define te elements on te reference domain te same definition can be used on a target element in te mes [8]. In order to define finite elements on parallelepiped we need te following polnomial spaces: Q ( ) {polnomials of maimum degree in m in ŷ and n in ẑ }. mn.. H ( Ω) conforming finite element spaces We start b defining te scalar spaces suitable for discretiing te potential [6]. Definition. Let. On te reference element te gradient conforming element is defined as follows: U ( ) Q ( ) were Q ( ) is te subspace of Q ( ) ecept constant multiple of te term and for. Ten we see te dimension of U ( ) is ( ) ( ). Definition. For an p U ( ) we define te following degrees of freedom: p ( a ) for te eigt vertices â of () pq ds for eac edges ê of q P ( e ) () e pq da for eac faces f of q Q ( ) f () f (4) pqda q Q ( )

5 AN ANALYSIS OF NEW FINITE ELEMENT SPACES 5 were Q ( f ) is te subspace of Q ( f ) ecept constant multiple of te term and Q ( ) is te subspace of Q ( ) ecept constant multiple of te term and for. Teorem. A scalar function p U ( ) is uniquel determined b te degrees of freedom ()-(4). And te finite element space U ( ) is conforming in H ( Ω ). Proof. Since te number of degrees of freedom and te dimension of U ( ) are bot it suffices to sow tat if p U ( ) and all te degrees of freedom ()-(4) of p vanis ten p. First we use te fact tat te verte degrees of freedom vanis on eac edge ê of f. For eample on te edge we ave p ( ) r for some r P ( e ). Coosing q r in te degrees of freedom () for tis edge sows tat r. Now using te fact tat p on eac edge ê of f wic we assume to be te face we ave p ( ) ( ) r for some r Q ( f ). Coosing q r in te degrees of freedom () sows tat r and ence p on f. B te same reason we see tat p on all faces. Tis proves te conformit in H ( Ω) space. And we ave p ( ) ( ) ( ) r. for some r Q ( ) Coosing q r in te degrees of freedom (4) proves r and ence p.

6 6 Remar 4. Our element as smaller number of degrees of freedom tan te well nown H ( Ω) conforming finite elements on parallelepiped. Te finite element space on a general element can be obtained b using te diagonal affine map F via U ( ) { p F p U ( )} in te usual wa. Ten we ave te following space: U { p H ( Ω) p U( ) for all τ }. (5) Using te degrees of freedom ()-(4) transformed on we can define an interpolation operator π : H ( ) U( ) b requiring te degrees of freedom of π p p vanis. Ten te global interpolation operator π is defined element-wise b for all τ. ( π p) π ( p ).. Te curl conforming finite element spaces We now present te edge elements wic is used to discretie te electric field [6]. Definition 5. Let. On te reference element te curl conforming element space V ( ) is defined b te subspace of Q ( ) ( ) ( Q Q ) were te elements in te set { αi j } j are replaced b te elements β and te tree elements γ i are replaced b te single element δ for i as follows: i

7 AN ANALYSIS OF NEW FINITE ELEMENT SPACES 7 β α α β α α β α α. δ γ γ γ Ten te dimension of ( ) V is ( )( ) ( ). To define te degrees of freedom we need two auiliar spaces. First we define ( ) curl Φ to be te subspace of ( ) ( ) Q Q were te two elements ( ) and ( ) are replaced b te single element ( ). To define te second space we use a replacement rule similar to te Definition 5. We define ( ) curl Ψ to be te subspace of ( ) ( ) ( ) Q Q Q were te elements { } φ j ij are replaced b te elements i v/ and te tree elements i ξ are replaced b te single element ζ for i as follows: / φ φ v

8 8 φ φ {( )} {( )} v/ {( )} φ φ {( )} {( )} v/ {( )} ξ {( )} ξ ξ {( )} ζ {( )} {( )}. Definition 6. For an v V ( ) te degrees of freedom are given on edge ê wit unit tangent t on faces f wit unit normal n and in te interior of as follows: v t q ds for e eac edges e of q P ( e ) (6) f ( v n ) q da for eac faces f of q curl ( Φ f ) (7) curl d ( ). v q q Ψ (8) Teorem 7. A vector function u V ( ) is uniquel determined b te degrees of freedom (6)-(8). And te finite element space V ( ) is conforming in H ( curl Ω). Proof. Since te number of conditions and te dimension of V ( ) are bot 5 7 it suffices to sow tat if all te conditions are ero ten u. First we consider te face. Ten te tangential component of û on tis face is ( u u ) Q ( )

9 AN ANALYSIS OF NEW FINITE ELEMENT SPACES 9 Q ( ). On eac edge of tis face te tangential component is polnomial of degree. From te degrees of freedom in (6) we see tat u t on eac edge. Tis implies tat on tis face we ave ( ) v u ( ) u v were ( v ) v ( ). Ten b coosing q v and q v in te degrees of freedom (7) we see v v. Hence u n on tis Φ curl face. B te same reason we see tat u n on all faces. Tis proves te conformit in H(curl) space. And we ave ( ) ( ) u w ( ) ( ) u w ( ) ( ) u w curl were ( w w w ) Ψ ( ). Coosing q ( w w w ) in degrees of freedom (8) we now tat u. Remar 8. Our element space as 5 fewer degrees of freedom tan te well nown Nedelec finite element space in eac element. Hence it is more efficient. For a generic element we define te finite element space on as T V( ) {( B ) v F v V ( )} in te usual wa. Ten we ave te following space: V { v H( curl Ω) v V( ) for all τ }. (9) Using te degrees of freedom (6)-(8) transformed on we can define te corresponding projection r : H ( ) V( )

10 for an arbitrar element. Ten te global projection operator r is defined piecewise b ( ) ( ) v r v r for all. τ.. Te divergence conforming finite element spaces In tis subsection we introduce divergence conforming finite element spaces wic will be used to discretie te magnetic induction [5]. Definition 9. Let. On te reference element te divergence conforming element space ( ) W is defined b te subspace of ( { ( )}) ( { ( )}) ( \ \ \ Q P Q P Q { ( )}) P were te vectors { } j ij a are replaced b i b for i as follows: b a a b a a. b a a Ten we see te dimension of ( ) W is ( ) ( ) ( ).

11 AN ANALYSIS OF NEW FINITE ELEMENT SPACES To define te degrees of freedom we need an auiliar space. We define ( ) to be te subspace of ( Q { P ( )}) Ψ div \ ( { ( )}) ( \{ \ Q P Q P ( )}) were vectors { ij } are replaced b v/ i for i as follows: φ j φ φ {( )} {( )} v / {( )} φ φ {( )} {( )} v / {( )} φ φ {( )} {( )} v / {( )}. div Note tat te definition of ( Ψ ) is similar to tat of W ( ) ecept tat te igest eponent is replaced b. Definition. For an u W ( ) te degrees of freedom are given on faces f wit unit normal n and in te interior of as follows: u n qda q Q ( f ) for eac face f () f div u q d ( ) q Ψ () were Q ( f ) Q ( ) is te subspace of Q ( ) \ { } for te face f in -plane etc.

12 Teorem. A vector function u W ( ) is uniquel determined b te degrees of freedom ()-(). And te finite element space W ( ) is conforming in H ( div Ω). Proof. Since te number of conditions and te dimension of W ( ) are bot 6{( ) } { ( ) } it suffices to sow tat if all te conditions are ero ten u. Since u n ( f ) it is clear Q tat () implies u n on eac face and tis proves te conformit in H(div) space. Also u ( u u u ) satisfies u ( ) v u ( ) v and u ( ) were v ( v v v ) Ψ div. Tus if we q v tae v in () we obtain u. Remar. Our element of inde lies between BDDF and RTN elements of inde. Altoug BDDF element as less degrees of freedom tan our element tere is no nown finite element corresponding to H(curl) space in relation to de Ram diagram. Our element as 6 ( ) fewer degrees of freedom tan te RTN element and we can construct a new H(curl) conforming element wic satisfies de Ram diagram. For a generic element we define te finite element space on as W ( ) { B ( ) ( w F w W )} det B in te usual wa. Ten we ave te following space: W { w H( div Ω) w W( ) for all τ }. () Using te degrees of freedom ()-() transformed on we can define te corresponding projection ω : H ( ) W( )

13 AN ANALYSIS OF NEW FINITE ELEMENT SPACES for an arbitrar element. Ten te global projection operator defined piecewise b ω is ω v ω ( v ) for all τ..4. An L ( Ω) conforming finite element spaces We will now define a new finite element space in L ( Ω) to complete te discrete de Ram comple. Definition. Let. On te reference element te L ( Ω) conforming element Z ( ) is defined as follows: ( Z ) Q ( ) were Q ( ) is te subspace of Q ( ) ecept constant multiple of te term and for. Ten we see te dimension of Z ( ) is ( ). Remar 4. Our new element as fewer degrees of freedom tan eisting L ( Ω) conforming finite elements used to complete te de Ram diagram. Hence using our finite elements we can effectivel calculate Mawell s equations. For a generic element we define te finite element space on as Z ( ) { q F q Z ( )} in te usual wa. Ten we ave te following space: Z { q L ( Ω) q Z( ) for all τ } () and define to be te L ( Ω) ortogonal projection into Z. P

14 4. Analsis for te Finite Element Spaces In tis section we will sow tat te finite element spaces U H ( Ω ) V H( curl Ω) W H( div Ω) and Z L ( Ω) are connected in an intimate wa b te following te discrete de Ram diagram [ ]: Also we will present error estimates of te interpolation operators π r ω and P. Teorem 5. If U is defined b (5) and In addition if p is sufficientl smoot suc tat defined ten we ave π p r p. V b (9) ten U V. r p and π p are Proof. Clearl if p U ten we see directl tat p V. Hence U V. To prove te commuting propert we map to te reference element and sow tat all degrees of freedom (6)-(8) vanis for π p r p. Ten we conclude tat π p r p. For te edge degrees of freedom (6) if t is tangent to [ a b] e and q P ( e ) ten using (6) and integration b parts we ave e ( π p r p ) t qds ( π p p ) t qds e

15 AN ANALYSIS OF NEW FINITE ELEMENT SPACES 5 ( π p p ) qds s e ( π )( ) ( π ) ( ) ( π ) q p p b p p a p p ds. s q Since P ( e ) s and using te verte interpolation propert and te degrees of freedom () for π we conclude tat te rigt-and side above vanises. For te face degrees of freedom we use te degrees of freedom in (7) togeter wit te divergence teorem in te plane containing f to sow tat if q Φ curl ( f ) ten f were ( π p r p ) n q da f f ( π p p ) q da e f f ( π p p ) n q ds ( π p p ) q da f f n f is te outward normal to f. Since n q P ( e ) f and ( f q Q f ) so tat rigt-and side vanises using te edge and face degrees of freedom () and () for π. We ave tus proved tat te face degrees of freedom (7) for π p and r p agree. Finall for te volume degrees of freedom we use te degrees of freedom in (8) togeter wit te integral identit to sow tat if curl q Φ ( ) ten

16 6 ( π p r p ) q d ( π p p ) q d ( π ) ( π ). p p q n da p p q d Since q n Q ( f ) for eac face f and q Q ( ) so te rigt-and side vanises using te face and volume degrees of freedom for π. Tis completes te proof. Teorem 6. If V is defined b (9) and W b () ten V W. In addition if u is smoot enoug suc tat r u and ω u are defined ten r u ω u. Proof. Te proof of V see []. To prove te commuting W propert we map to te reference element and sow tat te degrees of freedom given in () and () vanis for r u ω u. For te face degrees of freedom () we let f be a face of wit normal vector n and q Q ( f ). Let be te surface gradient. Using f te definition of projection operator ω te fact tat n ( v ) ( n v ) and integration b parts we ave f ( r u ω u ) n q da f ( r u u ) n q da f f f f ( n ( ru u )) q da

17 were AN ANALYSIS OF NEW FINITE ELEMENT SPACES 7 n ( ru u ) q da ( ( )) q ds f n f n ru u f n ( ru u ) q da ( ru u ) ( n n ) q ds f f f n is te unit normal vector to f in te plane containing te face f f. Since q curl ( f ) f Φ and q P ( e ) for eac edge ê of f te rigtand side of above formula vanises b te degrees of freedom (6) and (7). For te volume degrees of freedom () we let div q Ψ ( ). Using te definition of projection operator ω and Green s teorem of te following form: we ave u q d Ω ( r u ω u ) q d u Ω ( r u u ) q d f f q d n u q da Ω ( r u u ) q d ( n ( ru u )) q da. Since curl q Ψ ( ) and curl q Φ ( ) te rigt-and side of above formula vanises b te degrees of freedom (7) and (8). Now we will sow tat te L ( Ω) projection Po : L ( Ω) Z is related to ω. Teorem 7. If W is defined b () and Z b () ten W Z. For all sufficientl smoot functions we ave ω u u P o.

18 8 Proof. If u W ten we see directl tat u Z. Q For all q using te definition of P o and te Green s teorem we obtain ( ω up ) o u qdv ( ω u u ) qd V ( ) ( ) ω uu nq da ω uu q dv. Note tat q Q ( f ) for eac face f and q Ψ div ( ). So bot integrals on te rigt-and side vanis owing to te degrees of freedom () and (). Te above result and Teorems 5 and 6 sow tat te discrete de Ram comple commutes. Te following teorems provide error estimates for te projection operators wose proof is now standard [ 9 ]. Teorem 8. Let H τ be a regular famil of meses of Ω. If p ( Ω) and u H ( Ω ) ten tere is a constant C independent of suc tat p π p u r u u ω u C p C C u p P p C p. ( u u )

19 AN ANALYSIS OF NEW FINITE ELEMENT SPACES 9 Acnowledgement Tis wor was supported b te National Researc Foundation of orea grant funded b te Ministr of Education 7RCB References [] A. Alonso and A. Valli An optimal domain decomposition preconditioner for lowfrequenc time-armonic Mawell equations Mat. Comput. 68(6) (999) DOI: ttps://doi.org/.9/s [] A. Bossavit Computational Electromagnetism Academic Press San Diego 998. [] A. Bossavit Mied Finite Elements and te Comple of Witne Forms Academic Press London 988. [4] F. Brei J. Douglas R. Duran and M. Fortin Mied finite elements for second order elliptic problems in tree variables Numer. Mat. 5() (987) 7-5. DOI: ttps://doi.org/.7/bf9675 [5] J. H. im and Do Y. wa New curl conforming finite elements on parallelepiped Numer. Mat. () (5) DOI: ttps://doi.org/.7/s [6] J. H. im New H ( Ω) conforming finite elements on eaedra Int. J. Pure Appl. Mat. 9() (6) DOI: ttps://doi.org/.7/ijpam.v9i. [7] J. C. Nedelec Mied finite elements in R Numer. Mat. 5() (98) 5-4. DOI: ttps://doi.org/.7/bf9645 [8] Pilippe G. Ciarlet Te Finite Element Metod for Elliptic Problems Nort-Holland Publising Compan New Yor 978. [9] Peter Mon Finite Element Metods for Mawell s Equations Clarendon Press Oford. [] Peter Mon Analsis of a finite element metod for Mawell s equations SIAM J. Numer. Anal. 9() (99) DOI: ttps://doi.org/.7/7945 g