A new family of high regularity elements

Transcription

1 A new famly of hgh regularty elements Jguang Sun Abstract In ths paper, we propose a new famly of hgh regularty fnte element spaces. The global approxmaton spaces are obtaned n two steps. We frst buld an open cover of the computatonal doman and local approxmaton spaces on each patch of the cover. Then we construct partton of unty functons subordnate to the open cover dependng on the regularty requrement. The bass functons of the global space s gven by the products of the local bass functons and the correspondng partton of unty functons. The method can be used to construct fnte element spaces of any desred regularty. Approxmaton propertes and mplementaton detals are dscussed. Numercal examples for the bharmonc equaton are presented to show the effectveness of the proposed method. Keywords: hgh regularty fnte element, partton of unty, bharmonc equaton Introducton Hgh regularty fnte element spaces are of central mportance for the approxmaton of partal dfferental equatons of hgher order, for example, the Argyrs trangle [] and the Bogner-Fox-Schmt rectangle [] for the bharmonc equaton. However, classcal conformng fnte element spaces are rarely used n practce because they are dffcult to construct n general. We refer the readers to [, 6,,, 4, 5] for some efforts to construct hgh regularty fnte element spaces, n partcular, C elements. Alternatve methods, such as nonconformng fnte element methods [, 8, 7, 8, ] and mxed fnte element methods [7, 5, 9], are also proposed to treat hgher order problems. In ths paper, we propose a new famly of hgh regularty fnte element spaces. The man components are local approxmaton spaces on each patch of an open cover of the computatonal doman and the partton of unty functons. The global fnte element spaces nhert the approxmaton propertes of the local spaces and the smoothness of the partton of unty functons. The technque we used s the so-called partton of unty fnte element method (PUFEM) [4, 4]. We refer the readers to [, 9,, 3] and references theren for recent developments and applcatons of the PUFEM. Although ponted out n the fundamental paper of Melenk and Babuška [4], the ablty of the PUFEM to construct fnte element spaces of hgh regularty has not been fully explored to date. Ths s the motvaton of ths paper. To be precse, we apply a thn overlappng verson of the PUFEM (see [, 5, 6, 7]) to construct hgh regularty fnte element Department of Mathematcal Scences, Delaware State Unversty, Dover, DE 99, U.S.A. E-mal: jsun@desu.edu

2 spaces. The major advantages of the proposed method nclude () the smple choces of the local bass functons, for example, bquadratc polynomals n two dmenson, () the ablty to construct hgher regularty fnte element spaces by choosng adequate partton of unty functons, and () the easy extenson to hgher dmensons, for example, three dmensonal C elements. The paper s organzed as follows. In Secton, we frst ntroduce fundamental concepts and theores of the partton of unty method [4, 4]. Then we make necessary extensons n order to construct H conformng fnte element spaces. In Secton 3, we show examples of how to construct hgh regularty spaces n detal n one, two and three dmensons. Then we use these spaces to solve the bharmonc equaton n Secton 4. In Secton 5, we make conclusons and menton some future work. Partton of unty method In ths secton, we ntroduce the concept of the partton of unty fnte element method [4, 4] and make necessary extensons to facltate the constructon of H conformng fnte element spaces. We wll also remark on hgher regularty fnte element spaces. Defnton.. Let Ω R d be an open set, {Ω } be an open cover of Ω satsfyng a pontwse overlap condton: M N x Ω card { x Ω } M. Let {φ } be a partton of unty subordnate to the cover {Ω } satsfyng supp φ closure(ω ),, φ on Ω, φ L (R n ) C, φ L (R n ) C, d = dam Ω, d α φ x α C d, α =, L (R n ) where C, C, C are constants. Then {φ } s called an (M, C, C, C ) partton of unty subordnate to the cover {Ω }. The partton of unty {φ } s sad to be of degree m N f {φ } C m (R n ). The coverng sets Ω s are called patches. Remark.. The above condtons on the partton of unty functons are suffcent to construct H conformng fnte element spaces. To obtan hgher regularty fnte element spaces, we wll need addtonal condtons on α x αφ L (R n ) for α >. Defnton.. Let {Ω } be an open cover of Ω R d and let {φ } be a (M, C, C, C ) partton of unty subordnate to {Ω }. Let V be the approxmaton space on Ω. Then the space (.) V := { } φ V = φ v v V

3 s called the global approxmaton space. The space V s sad to be of degree m f V C m (Ω). The space V are referred to as the local approxmaton spaces. The approxmaton propertes of V depends on the local approxmaton spaces V and the partton of unty functons as we can see n the followng theorem. Theorem.. Let Ω R d be gven, and {φ }, {Ω }, {V } be as n the defntons above. Let u be the functon to be approxmated. Assume that the local approxmaton spaces V have the followng approxmaton propertes: on each patch Ω Ω, u can be approxmated by a functon v V such that Then the functon u v L (Ω Ω) ǫ (), (u v ) L (Ω Ω) ǫ (), D α (u v ) ǫ 3 (). α = L (Ω Ω) (.) u ap = φ v satsfes (.3) u u ap L (Ω) MC ( ǫ () ) / (.4) (.5) (u u ap ) L (Ω) M D α (u u ap ) α = L (Ω) ( [ C d ] ǫ () + C ǫ () ) / MN ( [ C d ] ǫ () + [ C d ] ǫ () + C ǫ 3() ) / where N s a constant dependng on the dmenson d. Proof. We wll show the proof of (.5). The proof of (.3) and (.4) can be found n [4]. Snce the functons φ form a partton of unty, we have u = φ u. Let x be the frst component of x R d. We have x 3 (u v) φ x L (Ω) = (u v ) x L (Ω) φ (u v ) L (Ω) φ (u v ) + x x L (Ω) (u v ) + 3 φ x L (Ω). 3

4 For any gven x Ω, there are at most M patches coverng t. Thus the sums φ x (u v ), φ x (u v ) x, and φ (u v ) x contan at most M terms for any fxed x Ω. We obtan 3 3M = 3M M φ x ( 3 φ (u v ) (u v ) x L (Ω) x L (Ω) (u v ) + M φ (u v ) L (Ω) x x L (Ω) +3M φ (u v ) x (u v ) + M φ (u v ) L (Ω Ω) x x L (Ω Ω) +3M φ (u v ) x ] [ ] ) ǫ () + C ǫ d () + 3C ǫ 3 (). φ x φ x [ C d φ (u v ) x L (Ω), L (Ω Ω) Then there exst a constat N dependng on the dmenson d and α (= ) such that D α (u u ap ) ( [ ] [ ] ) / C MN d ǫ () + C ǫ d () + C ǫ 3 (). L (Ω) α = L (Ω), Combnng the results n the above theorem, we obtan the followng estmate n H norm. Let u H (Ω) and u ap be defned as n (.) above. Then there exsts a constant C dependng on M, C, C, C, N such that (.6) u u ap H (Ω) C ( { d 4 + d } { + ǫ () + d ) + }ǫ () + ǫ3 (). In order to obtan a conformng H fnte element space, we need more restrctons on the open cover {Ω } and the local approxmaton spaces V s. Let u H k (Ω), k. We assume that (H) There exst two constants C, c > such that where h = max damω. ch damω Ch, for all, 4

5 (H) Each V has the followng approxmaton propertes: ǫ () Cd µ+ u Hk (Ω Ω ), ǫ () Cd µ+ u Hk (Ω Ω ), ǫ 3 () Cd µ u H k (Ω Ω ), for some µ >, the local approxmaton order. Usng the prevous theorem, t s straghtforward to show the followng theorem. Theorem.3. Assume the condtons n Theorem. and (H-H) hold. Then there exst constants C dependng on M, C, C and C such that (.7) (.8) (.9) u u ap L (Ω) Ch µ+ u Hk (Ω), (u u ap ) L (Ω) Ch µ+ u H (Ω), k D α (u u ap ) Ch µ u Hk (Ω). α = L (Ω) 3 Constructon of global approxmaton spaces Now we dscuss the detal of how to construct the hgh regularty conformng spaces. In partcular, we wll construct H conformng spaces. We wll also remark on the hgher regularty spaces. 3. One dmenson case Let Ω = (, ), n N, h = /n and defne x = h, =,,..., n, n +. Let Ω = (x, x + ) Ω. Obvously {Ω } s an open cover of Ω correspondng M = n Defnton.. On each patch Ω, we need to defne a local space V. In vew of Theorem.3, t s necessary to use at least quadratc bass functons for V f we try to solve fourth order problems, e.g., the bharmonc equaton. A quadratc bass on the reference element ˆΩ = (, ) s gven by (see Fg. ) v = x 3x +, v = 4x + 4x, v 3 = x x. For better approxmaton, we can choose hgher order polynomal bass, for example, the cubc bass or the Hermte bass on the reference element ˆΩ = (, ),.e., v = 9 x3 + 9x x +, v = 7 x3 45 x + 9x, v 3 = 7 x3 + 8x 9 x, v 4 = 9 x3 9 x + x, 5

6 or v = x 3 3x +, v = x 3 + 3x, v 3 = x 3 x + x, v 4 = x 3 x. Next we consder the partton of unty functons subordnate to {Ω }. The smplest choce of partton of unty functons mght be φ (x) = φ (x x ) where (3.) φ (x) = (x + h) (h x), x ( h, ], (h x) (h + x), x (, h), h 3, elsewhere. Other choces of partton of unty functons are possble, for example, (3.) φ (x) = + cos(πx/h), x ( h, h) and (3.) φ 3 (x) = h 5 (h + x) 3 (h 3hx + 6x ), x ( h, ], (h x) 3 (h + 3hx + 6x ), x (, h),, elsewhere. Assume that the local approxmaton spaces V s are gven by span {v,j, j =,...,N }. Also assume that the partton of unty functons are gven as above. Then the followng theorem shows that the global approxmaton space V s H (, ) conformng. Theorem 3.. Let V be the global approxmaton space constructed usng the local approxmaton spaces and partton of unty functons gven above. Then V H (, ). Proof. Let v V. Then φ v can be vewed as a functon defned on the whole doman Ω whch s a pecewse polynomal and a C (, ) functon. Hence any lnear combnaton of these functons s also a C (, ) functon whch mples V C (, ). By Theorem.. n [], we have that V H (, ). Remark 3.. To obtan hgher regularty global approxmaton spaces, we need to put more restrctons on the partton of unty functons and local approxmaton spaces. For example, to construct an H 3 conformng space, one could use the cubc bass or the Hermte bass for V (for convergence, we need polynomal bass of order 3 or hgher) and partton of unty functons φ 3 defned n (3.) (for regularty, we need that α x αφ L (R n ) C 3/d 3 for some constant C 3 and α = 3. See Remark..). 6

7 h h /8 /8 /8 h h h h Fgure : Left: Quadratc bass functons on Ω. Mddle: The partton of unty functon φ on Ω. Rght: The H conformng bass functons on Ω. 3. Two dmensonal rectangular meshes The above constructon can be extended to two and three dmenson cases easly. For smplcty, let Ω = (, ) (, ). We consder a unform overlappng rectangular mesh on Ω. Let n N, h = /n and defne Let x = h, =,,..., n, n +, y j = jh, j =,,..., n, n +. Ω,j = (x, x + ) (y j, y j+ ) Ω. Obvously {Ω,j } s an open cover of Ω. On each Ω,j, we need a local approxmaton space V,j. The choce of the local space s qute flexble as long as t can provde necessary local approxmaton ablty. For example, one may use the bquadratc bass functons for H conformng spaces. For the partton of unty functons n two dmenson, one can use the product of the partton of unty functons n the one dmenson case. For example, the partton of unty functons can be defned by φ,j (x, y) = φ (x x )φ j (y y j),.e., (see Fg. ) φ(x, y) = h 6 (x + h) (h x)(y + h) (h y), (x, y) ( h, ] ( h, ], (x + h) (h x)(h y) (h + y), (x, y) ( h, ] (, h), (h x) (h + x)(y + h) (h y), (x, y) (, h) ( h, ], (h x) (h + x)(h y) (h + y), (x, y) (, h) (, h),, elsewhere. As n one dmensonal case, the hgh regularty of the space can be acheved by a proper choce of the partton unty functons whle the approxmaton property reles on the local approxmaton spaces. For example, the global space V constructed from the b-quadratc local bass and the partton of unty functon φ,j gven above s H conformng. The proof s a straghtforward generalzaton of the one dmensonal case. 3.3 Three dmensonal rectangular meshes The constructon of V for three dmenson case s smlar. Let Ω = (, ) (, ) (, ). We consder a unform rectangular mesh on Ω. Let n N, 7

8 Ω,j+ Ω +,j+ Ω o Ω,j Ω +,j Fgure : Left: Overlappng rectangular mesh n D. Ω o = Ω,j Ω,j+ Ω +,j Ω +,j+. Note that every pont x n the doman s covered by 4 patches,.e., M = 4 n Defnton.). Rght: The partton of unty functons on Ω o. h = /n and defne x = h, =,,..., n, n +, y j = jh, j =,,..., n, n +, z k = kh, k =,,..., n, n +. Let Ω,j,k = (x, x + ) (y j, y j+ ) (z k, z k+ ) Ω. Then {Ω,j,k } s an open cover of Ω. As above, one may use the smple trquadratc polynomals as local bass. For the partton of unty functons, one can smply use where φ = h 9 φ,j,k (x, y, z) = φ(x x, y y j, z z k ) (x + h) (h x)(y + h) (h y)(z + h) (h z), ( h, ] ( h, ] ( h, ], (x + h) (h x)(y + h) (h y)(h z) (h + z), ( h, ] ( h, ] (, h), (x + h) (h x)(h y) (h + y)(z + h) (h z), ( h, ] (, h) ( h, ], (x + h) (h x)(h y) (h + y)(h z) (h + z), ( h, ] (, h) (, h), (h x) (h + x)(y + h) (h y)(z + h) (h z), (, h) ( h, ] ( h, ], (h x) (h + x)(y + h) (h y)(h z) (h + z), (, h) ( h, ] (, h), (h x) (h + x)(h y) (h + y)(z + h) (h z), (, h) (, h) ( h, ], (h x) (h + x)(h y) (h + y)(h z) (h + z), (, h) (, h) (, h),, elsewhere. As above, one can easly construct a H conformng global space usng the above partton of unty functons and the tr-quadratc polynomal local bass. 3.4 Two dmensonal case on trangular meshes Now we dscuss how to construct H conformng spaces based on gven trangular meshes. Let T be a trangular mesh for Ω. We frst construct an open cover {Ω }. Our constructon s assocated to the nodes of T. Let be a node of 8

9 Fgure 3: Left: A trangular mesh of Ω. Rght: A polygon patch Ω assocated wth vertex. T. The unon of all trangles whose vertces nclude gves a polygon, denoted by Ω. Let N be the ndex set of all nodes n T. Then {Ω }, N s an open cover of Ω. For example, Fg. 3 shows a trangular mesh. Then Ω s the polygon whch s the unon of trangles (,, 5), (, 5, 4), (, 4, 6), (, 6, 3), (, 3, ). It s obvous that {Ω }, =,..., s an open cover of Ω. Remark 3.3. Other types of constructon are possble. For example, we can assocate the open cover wth trangles n the orgnal mesh. The patch Ω s the unon of a trangle and all other trangles surroundng t. Now we need to defne a local approxmaton space V on Ω. Snce Ω s the unon of trangles, one may choose V as the space of contnuous pecewse quadratc functons wth respect to the trangulaton of Ω. Remark 3.4. Polygonal fnte element nterpolants usng ratonal polynomals for convex polygons was dscussed by Wachspress [3]. We refer the readers to [] and references theren for recent developments n the constructon of fnte element nterpolants on polygonal domans. The partton of unty functons on Ω can be defned smlarly. We frst construct a proper functon on each trangle and the unon of all these functons wll gve the correspondng partton of unty functons. For example, on the reference trangle ˆK = {(, ), (, ), (, )}, the proper functon for Ω (,) can be defned as φ ˆK = x 3 y 3 + 5x 4 3x y + 5y 4 6x 5 + 3x 3 y + 3x y 3 6y 5. The node (, ) s the center of Ω (,) and {(, ), (, )} s one edge of t. At (, ), φ {(,),(,),(,)} s one and all ts frst order and second order partal dervatves vansh. At (, ) and (, ), the value and all frst order and second order partal dervatves of φ {(,),(,),(,)} vansh. Fg. 4 shows the partton of unty functon on the patch Ω shown n Fg. 3. 9

10 Fgure 4: The partton of unty functon on the polygonal patch Ω. 4 Applcaton to the bharmonc equaton We consder the followng homogeneous Drchlet problem for the bharmonc equaton (4.3a) (4.3b) u = f n Ω, u = ν u = on Γ, where Γ = Ω and ν s the unt outward normal to Γ. The weak formulaton s to fnd u H(Ω) such that (4.4) a(u, v) := u v dx = fv dx =: (f, v), v H(Ω). Ω Snce the blnear form a(u, v) s H (Ω)-ellptc, there exsts a unque soluton to (4.4) []. The correspondng dscrete problem can be stated as to fnd u h V H(Ω) such that (4.5) u h v dx = fv dx, v V Ω where V s the global approxmaton space constructed n the prevous secton. Theorem 4.. Assume that V s the global approxmaton space obtaned as n (.) and condtons n Theorem. hold. Let u H (Ω) and u h V be the solutons of (4.4) and (4.5), respectvely. Then there exsts a constant C such that (4.6) ( { u u h,ω C d 4 + d Ω Ω } { + ǫ () + d ) / + }ǫ () + ǫ3 () Proof. Snce the blnear form a s ellptc, we have, by Céa s lemma, Now u ap V and t s obvous that Thus (4.6) holds. u u h,ω C nf v h V u v h H (Ω). nf v h V u v h H (Ω) u u ap H (Ω).

11 Combnng Theorem.3 wth Theorem 4., we obtan the followng theorem for one dmenson case correspondng to the example we dscuss above. Theorem 4.. Assume that u H (Ω) and (H) holds for µ and k 3. Then there exsts a constant C such that (4.7) u u h,ω Ch µ / u Hk (Ω). The followng theorem gves the L estmate. Theorem 4.3. Assume that u H (Ω) and (H) holds for µ and k 3. Then there exsts a constant C such that (4.8) u u h,ω Ch µ u Hk (Ω). Proof. The theorem can be proved usng Ntsche s trck whch s smlar to the L estmate for Posson equaton wth homogeneous Drchlet boundary condton (for example, see [3]). Thus we omt the detals. 4. One dmensonal examples In the followng we wll show some numercal results for the one dmensonal case. Let Ω = (, ), we have (4.9) (4.) u (4) = f, on (, ) u() = u() = u () = u () =. Let u = sn (πx) whch satsfes the bharmonc equaton (4.9) and (4.) wth f(x) = 8π 4 cos(πx). We use quadratc local bass and partton of unty functons gven by (3.), (3.) and (3.). The errors of the numercal soluton are shown n Fg. 5, Fg. 6 and Fg. 7, respectvely. All these three examples, the error convergence rate s O(h) n L norm and O(h / ) n the H sem-norm. Next we let u(x) = x 6 3x 5 + 3x 4 x 3. It s easy to check that u satsfes the bharmonc equaton (4.9) and (4.) wth f(x) = 36x 36x + 7. We frst use the quadratc bass functon on each Ω. The partton of unty functon s gven by (3.). We show the error n the L norm and the H sem-norm of the numercal soluton n Fg. 8. It can be seen that the convergence rate s O(h ) n the L norm and O(h) n the H sem-norm. Then we use Hermte bass functons on each Ω and keep the same partton of unty functons. The numercal results are shown n Fg. 9. The error convergence rate s O(h 4 ) n the L norm and O(h ) n the H sem-norm. For ths partcular example, we obtan better convergence rates. 4. A two dmensonal example Now we consder a two dmensonal example on a rectangular mesh. We choose u = sn (πx)sn (πy) such that f(x, y) = 4π 4 4π 4 cos (πy) 4π 4 cos (πx) + 64π 4 cos (πx)cos (πy). We use the H conformng fnte element space descrbed n Secton 3.. In Fg., we plot the numercal result. The error convergence rate s O(h ) n the L norm and O(h) n the H sem-norm.

12 .5 L error H error (Error) (N) Fgure 5: The error of the numercal soluton n log scale. The exact soluton s gven by u(x) = sn (πx). Local bass are quadratc bass functons and the partton of unty functons are gven by (3.). The error convergence rate s O(h) n the L norm and O(h / ) n the H sem-norm..5 L error H error (Error) (N) Fgure 6: The error of the numercal soluton of n log scale. The exact soluton s gven by u(x) = sn (πx). Local bass are quadratc bass functons and the partton of unty functons are gven by (3.). The error convergence rate s O(h) n the L norm and O(h / ) n the H sem-norm. 5 Conclusons and future work In ths paper, we propose a new famly of hgh regularty fnte element spaces. Based on an overlappng mesh of the computaton doman, the global approxmaton spaces are obtaned by choosng the local approxmatons spaces and the partton of unty functons approprately. The major advantage les n the smplcty and effcency of the method for hgher dmensonal problems. For

13 L error H error (Error) (N) Fgure 7: The error of the numercal soluton of u h n log scale. The exact soluton s gven by u(x) = sn (πx). Local bass are quadratc bass functons and the partton of unty functons are gven by (3.). The error convergence rate s O(h) n the L norm and O(h / ) n the H sem-norm. L error H error 3 (Error) (N) Fgure 8: The error of the numercal soluton of u h n log scale. The exact soluton s gven by u(x) = x 6 3x 5 + 3x 4 x 3. Local bass are quadratc functons and the partton of unty functons are gven by (3.). The error convergence rate s O(h ) n the L norm and O(h) n the H sem-norm. example, n the D case of rectangular meshes, the local degree of freedom s 9 for the proposed method comparng to 6 for the Bogner-Fox-Schmt rectangle. Whle ths does not necessarly mples less degrees of freedom globally snce we use an overlappng mesh, the proposed method lead smple mplementaton and potental savngs for hgher dmensonal problems. For hgher regularty elements (H k, k 3, conformng elements), the proposed method s smpler and wll lead much less degrees of freedom snce the possble use of normal vectors 3

14 L error H error 3 4 (Error) (N) Fgure 9: The error of the numercal soluton of u h n log scale. The exact soluton s gven by u(x) = x 6 3x 5 + 3x 4 x 3. Local bass are Hermte bass functons and the partton of unty functons are gven by (3.). The error convergence rate s O(h 4 ) n the L norm and O(h ) n the H sem-norm. to defne degrees of freedom s avoded. Note that the degrees of freedom for normal vectors are not respected by affne transformatons n general. Furthermore, the proposed method nherted the major advantages of the PUFEM such as the ablty to use dfferent local bass functons to address local behavor of the soluton whch s known as an a pror. Future work ncludes the applcaton of the method on trangle and tetrahedron meshes n two and three dmenson, and the mplementaton of essental boundary condtons. References [] J.H. Argyrs, I. Fred and D.W. Scharpf, The TUBA famly of plate elements for the matrx dsplacement method. Aero. J. Roy. Aero. Soc. Vol. 7 (968), [] I. Babuska, U. Banerjee and J.E. Osborn, Survey of meshless and generalzed fnte element methods: a unfed approach. Acta Numerca, 3, 5. [3] I. Babuska, U. Banerjee, J.E. Osborn, and Q. L, Quadrature for meshless methods, Int. J. Numer. Meth. Engng., Vol. 76 (8), [4] I. Babuška and J.M. Melenk, The partton of unty method, Internat. J. Numer. Methods Engrg., Vol. 4 (997), [5] C. Bacuta and J. Sun, Partton of unty fnte element method mplementaton for Posson equaton, Advances n Appled and Computatonal Mathematcs, Nova Publshers, 6,

15 .5 L error H error (Error) (N) Fgure : Left: The plot of the numercal soluton (h = /3). Rght: The error of the numercal soluton n log scale. The error convergence rate s O(h ) n the L norm and O(h) n the H sem-norm. [6] C. Bacuta and J. Sun, Notes on the Schwarz alternatng method for partton of unty FEM, Dyn. Contn. Dscrete Impuls. Syst. Ser. A Math. Anal., Vol. 6 (9), Dfferental Equatons and Dynamcal Systems, suppl. S, 5. [7] C. Bacuta, J. Sun and C. Zheng, Partton of unty refnement for local approxmaton, Numercal Methods for Partal Dfferental Equatons, n press. [8] S.C. Brenner and L.Y. Sung, C nteror penalty methods for fourth order ellptc boundary value problems on polygonal domans. J. Sc. Comput., Vol. /3 (5), [9] F. Brezz and M. Fortn, Mxed and Hybrd Fnte Element Methods. Sprnger Seres n Computatonal Mathematcs, Vol. 5. New York, Sprnger, 99. [] P.G. Carlet, The Fnte Element Method for Ellptc Problems, Classcs n Appled Mathematcs, 4, SIAM, Phladelpha,. [] E.H. Georgouls and P. Houston, Dscontnuous Galerkn methods for the bharmonc problem, IMA J. Numer. Anal. Vol. 9 (9), [7] R.S. Falk, Approxmaton of the bharmonc equaton by a mxed fnte element method, SIAM J. Numer. Anal. Vol. 5 (978), No. 3, [] Y. Huang and J. Xu, A conformng fnte element method for overlappng and nonmatchng grds, Math. Comput., Vol. 7 (), No. 43, [3] W. Hackbusch, Ellptc Dfferental Equatons, Theory and Numercal Treatment, Sprnger-Verlag, 99. [4] J.M. Melenk and I. Babuška, The partton of unty fnte element method: Basc theory and applcatons, Comput. Methods Appl. Mech. Engrg., Vol. 39 (996),

16 [5] P. Monk, A mxed fnte element method for the bharmonc equaton, SIAM J. Numer. Anal. Vol. 4 (987), No. 4, [6] J. Morgan and L.R. Scott, A nodal bass for C pecewse polynomals of degree n. Math. Comp. Vol. 9 (975), [7] I. Mozolevsk and E. Sul, A pror error analyss for the hp-verson of the dscontnuous Galerkn fnte element method for the bharmonc equaton. Comput. Methods Appl. Math., Vol. 3 (3), (electronc). [8] I. Mozolevsk and E. Sul, and P. Bosng, hp-verson a pror error analyss of nteror penalty dscontnuous Galerkn fnte element approxmatons to the bharmonc equaton. J. Sc. Comput., Vol. 3 (8), [9] H.S. Oh, J.G. Km and W.T. Hong, The pecewse polynomal partton of unty functons for GFEM, Comput. Method Appl. Mech. Engrg. Vol. 97 (8), [] P. Oswald, Herarchcal conformng fnte element methods for the bharmonc equaton. SIAM J. Numer. Anal. Vol. 9 (99), [] T. Stroubouls, I. Babuška, R. Hdajat, The generalzed fnte element method for Helmholtz equaton: theory, computaton, and open problems, Comput. Methods Appl. Mech. Eng., Vol. 95 (6), [] N. Sukumar and E.A. Malsch, Recent Advances n the Constructon of Polygonal Fnte Element Interpolants, Arch. Comput. Meth. Engng. Vol. 3 (6), No., [3] E.L. Wachspress, A Ratonal Fnte Element Bass. Academc Press, New York, N.Y [4] M. Wang and J. Xu, The Morley element for fourth order ellptc equatons n any dmensons. Numer. Math. Vol. 3 (6), [5] S. Zhang, A C -P fnte element wthout nodal bass. MAN Math. Model. Numer. Anal. Vol. 4 (8), No.,