Te Fnte Element Metod: A Sort ntroducton Wat s FEM? Te Fnte Element Metod (FEM) ntroduced by engneers n late 50 s and 60 s s a numercal tecnque for solvng problems wc are descrbed by Ordnary Dfferental Equatons (ODE) /Partal Dfferental Equatons (PDE) wt approprate boundary/ntal condtons or to solve problems tat can be formulated as a functonal mnmzaton. Wy FEM? Greater flexblty to model complex geometres. Can andle general boundary condtons. Varable materal propertes can be andled. Clear structure and versatlty elps to construct general purpose software for applcatons. Has a sold teoretcal foundaton wc gves added relablty and makes t possble to matematcally analyze and estmate te error n te approxmate soluton. We now descrbe te mportant steps of classcal/standard fnte element metod. We start wt a boundary value problem n te classcal formulaton. After defnng an approprate varatonal formulaton for te bounday value problem, we gve a fnte element formulaton. Step : We start wt te followng two pont boundary value problem. (P C ) : Gven f fnd u suc tat Au (a(x)u ) + a 0 (x)u f n (0, ), u(0) u() 0. Te followng assumptons on te coeffcents are made: a(x) and a 0 (x) are smoot functons wt a(x) α 0 > 0, a 0 (x) 0 n Ī, f L2 (), max x Ī( a(x), a 0 (x) ) M. Step 2: Galerkn Varatonal Formulaton (Weak Formulaton) For a boundary value problem defned usng an operator of order 2m, admssble space s cosen as H m (Ω) along wt all essental boundary condtons.e., boundary condtons nvolvng dervatves of u of order m. n ts case, we coose te admssble space as V H 0 (). Multplyng () by v V, ntegratng left and sde of () by parts and usng te boundary condtons, we get: were (P G ) : Gven f n L2 (), fnd u n V suc tat a(u, v) l(v) v V, a(, ) : V V R s a symmetrc, contnuous, blnear form defned by: a(u, v) (a(x)u v + a 0 (x)uv) dx u, v V (3) () (2)
l( ) : V R s a contnuous lnear form defned by: l(v) fv dx v V. (4) Lemma: (P G ) s well-posed. Step (2) : Rtz Varatonal Problem Gven f n L 2 (), fnd u n V suc tat J(u) nf (P R ) : J(v) v V were J(v) 2a(v, v) l(v) v V, a(, ) and l( ) are defned by (3) and (4) respectvely. (5) Lemma: (P G ) and (P R ) are equvalent. Step 3: Galerkn Fnte Element Problem Frstly, we construct a fnte dmensonal subspace V of V. For any postve nteger N +, let {0 x 0 < x <... < x N+ } be a partton of Ī nto subntervals (fnte elements) j (x j, x j ), j N +, wt lengt j x j x j. parameter assocated wt te mes. Te dscrete soluton wll be sougt n V defned by: V s a subspace of V. max j s te j N+ V {v : v C 0 (Ī), v j P ( j ), j N +, v (0) v () 0} (6) V s fnte dmensonal, snce on eac subnterval j, v P ( j );.e., we ave 2(N + ) degrees of freedom. Contnuty constrants at te nodal ponts x,..., x N and boundary condtons v (0) v () 0 mply dmv 2(N + ) (N) 2 N. Remark: Ts s te smplest coce for V. Oter coces are possble. Te Galerkn fnte element problem (PG ) correspondng to (P G) can be defned as: (PG) : Gven f n L2 (), fnd u n V V suc tat a(u, v ) l(v ) u V..e., ( a(x) du dx ) dv dx + a 0(x)u v dx fv dx v V. ) can be defned as: Te Rtz fnte element problem (PR (P R) : were J(v ) 2 a(v, v ) l(v ). Gven f n L2 (), fnd u n V V suc tat J(u ) nf v V J(v ), Usng Lax-Mlgram lemma, (PG ) s well-posed. Step 4: Constructon of Canoncal Bass Functons for V. (7) (8) 2
dmv N. Te bass functons for V can be cosen to be te Lagrange polynomals {φ }N wc satsfy φ (x j) δ j, j N, were wen j δ j 0 wen j.e., φ (x) x x n [, x ] x x + x x + n [x, x + ] 0, oterwse. Note tat supp φ [, x + ]. For v V, v (x) β φ (x), v (x j ) β φ j (x j) β j. Hence β j are notng but values of v at te nodal ponts..e., v (x) v (x )φ (x) (9) Te unknown functon u V can also be expressed as wt (u ) N (u (x )) N. u u (x )φ (x) u φ (x) (0) Step 5: Reducton of (PG ) nto Matrx Equatons. From (7), a(u, v ) l(v ) v V and from (0), u ( N ).e., a u φ, φj l(φ j ) j N. u a(φ, φ j ) l(φj ) j N. u φ KU F were K [a(φ, φj )],j N, U (u ) N, F (l(φ j )) j N. Propertes of K: () K s sparse: a(φ, φj ) 0 wenever Supp φ and Supp φj does not ntersect. Snce Supp φ [, x + ] n ts case, we observe tat K s trdagonal. (2) K s symmetrc: Ts s because te blnear form a(, ) s symmetrc. (3) K s postve-defnte: U T KU j u u j a(φ, φ j ) a u φ, u j φ j j a(u, u ) α u 2 V > 0. 3
K s postve-defnte. Step 6: Constructon of K, F for mplementaton purpose On [, x + ], we ave x n [, x ] x φ x x + n [x, x + ] x x + 0 oterwse. x x 2 n [x 2, ] x 2 φ x x n [, x ] x 0 oterwse. x x n [x, x + ] x + x φ + x x +2 n [x +, x +2 ] x + x +2 0 oterwse. For 2,..., N, a, a(φ, φ ) a(x) [ 2 a, 2 a(x)dx + dφ dx dφ dx dφ + dx dx + a 0 (x) n [, x ] n [x, x + ] 0 oterwse. n [x 2, ] n [, x ] 0 elsewere. n [x, x + ] + n [x +, x +2 ] +2 0 elsewere. ( ) ( x x x x a 0 (x)(x )(x x)dx ] ) dx [ a, a(φ, φ ) x ] x 2 a(x)dx + a 0 (x)(x ) 2 dx [ x+ + ] + 2 a(x)dx + a 0 (x)(x x + ) 2 dx + x x [ + + ] a,+ a(φ, φ+ ) a(x)dx + a 2 0 (x)(x x + )(x x)dx + x x a, a 2, a NN, a N,N are to be also computed. For F ( l(φ )), j N j+ j ( ) xj+ ( ) f j l(φ j f, ) φ j fφ j dx xj x xj+ f dx + f dx. x j x j j x j j+ Remarks:. Te fnte element metod (wt constant mes sze) concdes wt fnte dfference metod except for te fact tat an average of f over (x j, x j+ ) s now used nstead of pont values of f(x j ). 2. FE metod s tus Galerkn metod appled to a specal coce of fnte dmensonal subspace, namely contnuous pecewse lnear elements n ts case. Step 7: Error Estmates We need te followng results for obtanng te estmates u u V, u u 0,. Cea s Lemma: Let u and u be te solutons of (P G ) and (PG ) respectvely. Ten u u V C nf v V u u V () 4
Proof: From (2) and (7), we get v V, a(u, v ) l(v ) a(u, v ) l(v ) a(u u, v ) 0 v V. (2) ( (2) s called Galerkn ortogonalty property,.e., te fnte element soluton u s te ortogonal projecton of te exact soluton u onto V wt respect to te nner product a(, ) ). α u u 2 V a(u u, u v ) (By coercvty of a(, )) a(u u, u v ) + a(u u, v u ) }{{} 0 (usng (2)).e., u u 2 V M α u u V u v V (Boundedness of a(, )). n case u u, te result s trval. For u u, u u V M α u v V v V. u u V C nf v V u v V wt C M α. Remark: u s te best approxmaton of u n V w.r.t a a(,. nterpolaton Operator For a functon v C 0 ( Ω), wt v(0) v() 0, we defne te nterpolant operator Π : v H 0(0, ) C 0 (0, ) V, (Π v)(x ) v (x ) v(x ) 0 (N + ). We are now nterested to fnd u u 0 and u u so tat we can use ts n () to estmate u u V. Approxmaton Propertes of nterpolaton operator ( Assume tat u H 2 /2 (). We know u 2 0 u (x) dx) 2 s a sem-norm. By Sobolev mbeddng teorem, u C (). Applyng mean-value teorem to u [, x ], we get (, x ) suc tat n ts nterval, u (x) u(x ) u( ) u (). Hence, n ts nterval, we ave u () u(x ) u( ) u (x) u (x) u (x) u () u (t)dt 5
Applyng Caucy-Scwarz nequalty, we ave u (x) u (x).u (t)dt ( ( 2 /2.e., u (x) u (x) /2 To derve a bound on u(x) u (x) : u(x) u (x) u(x) u (x) ( ( [u (t) u (t)] dt (u (t) u (t)) dt u (t) u (t) dt u (t) 2 dt x (, x ) x (, x ). (3) /2 ( dx (usng (3)).e., u(x) u (x) 3/2 ( x (, x ). (4) Squarng (3) and (4), ( ntegratng and summng over all elements and takng square roots, we get ) x u (x) u (x) 2 n ( x ) x u (x) u (x) 2 dx dx u u 2, 2 u 2 2, Error Estmates, were max N. u u, u 2,. (5) Smlarly, we obtan u u 0, 2 u 2, from (4). (6) () Frst we fnd an estmate for u u V u u,. u u V C u u V (usng ()) u u 2 V u u 2 0, + u u 2, 4 u 2 2, + 2 u 2 2, ( 4 + 2 ) u 2 2, 2 ( + 2 ) u 2 2, u u 2 V 22 u 2 2, (usng < lengt() )... u u V C u 2, u H 2 () H0 (). (7) 6
(2) An estmate for u u 0, s found usng Aubn-Ntsce s dualty argument. Consder te followng adjont ellptc problem: For g L 2 (), let φ be te soluton of (a(x)φ ) + a 0 (x)φ g, x, (8) φ(0) φ() 0 Ts problem satsfes te regularty result φ 2 C g. (9) Multplyng (8) by e u u and ntegratng over, we get a(e, φ) (e, g) (e, g) a(e, φ φ ) (usng (2)).e., (e, g) C e V φ φ V C e V g 0, (usng (9)) Te result follows by coosng g e..e., e 2 0, C e V e 0, u u 0, C 2 u 2, Remark: () Te analyss could ave been done for a more general fnte element space consstng of polynomals of degree r were r 2. (Te case dscussed was for r 2). Ten, we would obtan u u, C r u r, and u u 0, C r u r, under te assumpton of addtonal regularty u H r (). (2) Computatonal Order of Convergence : For 0 < 2 <, let u u C(u) α u u 2 C(u) α 2 ( ) u u Ten te order of convergence s α log / log( ) u u 2 2 n te absence of exact soluton, u may be replaced by a more refned computed soluton. 7