Approximation of functions by piecewise defined trial functions

Transcription

1 Approiation of functions by picwis fin tria functions In a prvious chaptr was approit by a continuous function fin ovr th who oain R with ˆ ψ + { ;,2,3, L} ψ Γ n Γ Γ R a n its bounary such that anψ fin gobay ovr coptn ss rquirn t 0 Main rawback : tratnt of arbitrary gotris bounary conitions to b ipos Γ

2 ow Γ W R E W R Γ Approiation of functions by picwis fin tria functions bounary of ; ach { } is approiat picwis U Γ E E Γ W arbitrary,but ay bsip ; W R R I Γ E Γ Γ Γ pty st ovr ach suboain ocay nough

3 Approiation of functions by picwis fin tria functions Howvr picwis approiation of triafunction proucs a oss of rguarity for th gobay assb functions. In D th nts arintrvas

4 Approiation of functions by picwis fin tria functions

5 So typica ocay fin narrow-bas shap functions Approiat a iscrtiz by a 0; ˆ U M ψ n + M L n U M n continuous st of [, ] points function fin { ;,2, KM } ovr with with ˆ picwis constant n [ 0, L ]

6 Approiating a givn function in D

7 Approiating a givn function in D

8 Point coocation & picwis constant approiation of D functions s Ej_3_ routin functions tria by orthonoraity of 0 ˆ 0 with Mtho WRM Wight Rsiua Using M f K R W n δ δ δ δ δ ψ

9 Garkin & picwis constant approiation of D functions s Ej_3 2 routin Using W K f R Wight W Rsiua ˆ δ Mtho WRM M n by orthogonaity of 0 tria functions Point coocation a [ ] Garkin soution a [ ]

10 Point coocation & picwis inar approiation of D functions s Ej_3 3 routin Using W th K f Ka f Wight unknowns R δ δ Rsiua Mtho WRM with ψ 0 M n ˆ δ δ 0 ar now noa vaus insta of nt - wis vaus is th sa as using δ by orthonora ity of point coocation with picwis constant! tria functions

11 picwis constant vs inar

12 Garkin & picwis inar approiation of D functions nos in D naturanubring of * assuing a for inar * si 0 0 ˆ Mtho WRM Using Wight Rsiua f - K W R W M n

13 Garkin & picwis inar intgra krn for Garkin K zros+,+; f zros+,; for 2: psi 0:2'/2; psi*--+-; a shap_ej_3 4,-,; b shap_ej_3 4,,-; K,- K,- + trapz,b.*a; K, K, + trapz,b.*b; psi*+-+; a shap_ej_3 4,+,; b shap_ej_3 4,,+; K, K, + trapz,a.*a; K,+ K,+ + trapz,a.*b; psi 0:20'/20; psi*--+-; f, f, + trapz,shap_ej_3 4,-,.*ffun_Ej_3 4; psi*+-+; f, f, + trapz,shap_ej_3 4,+,.*ffun_Ej_3 4; n

14 +; psi 0:2'/2; psi*--+-; a shap_ej_3 4,-,; b shap_ej_3 4,,-; K,- K,- + trapz,b.*a; K, K, + trapz,b.*b; ; psi*+-+; a shap_ej_3 4,+,; b shap_ej_3 4,,+; K, K, + trapz,a.*a; K,+ K,+ + trapz,a.*b; Garkin & picwis inar bounary trs +; psi 0:20'/20; psi*--+-; f, f, + trapz,shap_ej_3 4,-,.*ffun_Ej_3 4; ; psi*+-+; f, f, + trapz,shap_ej_3 4,+,.*ffun_Ej_3 4;

15 picwis inar Garkin rsuts

16 Garkin & picwis inar Ent-wis assbing for k:nu psi 0:0'/0; no iconk,; no2 iconk,2; psi*nono2,-nono,+nono,; a shap_ej_3 4,nono2,,nono,; b shap_ej_3 4,nono,,nono2,; Kk,, Kk,, + trapz,a.*a; Kk,,2 Kk,,2 + trapz,a.*b; Kk,2, Kk,2, + trapz,b.*a; Kk,2,2 Kk,2,2 + trapz,b.*b; fk, fk, + trapz,a.*ffun_ej_3 4; fk,2 fk,2 + trapz,b.*ffun_ej_3 4; n % gathr K an f in Kg an fg Kg zros+,+; fg zros+,; for k:nu no iconk,; no2 iconk,2; Kgno,noKgno,no+Kk,,; Kgno,no2Kgno,no2+Kk,,2; Kgno2,noKgno2,no+Kk,2,; Kgno2,no2Kgno2,no2+Kk,2,2; fgno, fgno, + fk,; fgno2, fgno2, + fk,2; n % sovr a Kg\fg; icon [ ]

17 Approiating a givn function in 2D Using picwis constant on triangs Fig 3.2 a

18 Approiating a givn function in 2D Using picwis inar on triangs Fig 3.2 b