' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult - PDF document

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Higher o rder multilevel �nite element metho ds based on extrap olation quadrature Ulrich R� ude Theo ry Institute on Numerical Quadrature August 22 { 24, 1994; Argonne, IL � � Titel 0.1 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult � at f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Problem Many physical p roblems can b e stated as va ria- tional minimizati on p roblems of the fo rm min E ( u ) fo r u 2 U where U is a function space and and E ( u ) involves integrals. � � Problem 1.1 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Example 1 1 U = H (0 ; 1) 0 and Z � � 1 2 0 E ( u ) = d ( x ) u ( x ) � 2 f ( x ) u ( x ) dx: 0 This leads to the t w o p oint b ounda ry value p rob- lem 0 0 ( d ( x ) u ) = f in (0 ; 1) ; u (0) = u (1) = 0 : � � Examples 2.1 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Example 2 ! Z � � 1 4 2 u 0 2 E ( u ) = u ( x ) � c � u dx: 2 0 This leads to the t w o p oint BVP 00 3 u + c ( u � u ) = 0 in (0 ; 1) ; u (0) = � 1 + �; u (1) = 1 � �: � � Examples 2.2 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Example 3 1 2 U = H (�) ; � � R : 0 Z Z 2 E ( u ) = ( r u ) � 2 f u dxdy � This b ecomes Laplace's equation � u = f in � u = 0 on @ � � � Examples 2.3 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Example 4 Equations of elasticit y 1 1 U = H (�) � H (�) " Z Z ~ @ u @ u @ u @ u E 1 1 2 2 E ( u ) = + 1 + � @ x @ x @ x @ x � 1 1 2 2 � + div u div u 1 � � # @ u @ u @ u @ u 1 1 2 1 2 + ( + ) ( + ) dxdy 2 @ x @ x @ x @ x 2 1 2 1 Z + g u + g u ds; 2 ; 1 1 2 ; 2 2 � N ~ where E is Y oung's elasticit y mo dulus � is the P oisson ratio T g = ( g ; g ) a re the surface tractions 2 2 ; 1 2 ; 2 � � Examples 2.4 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Example 4 cont'd �: F ? ? ? ? ? ? ? ? ? ? ? @ � @ � @ � @ � @ � @ � @ � @ � @ � � @ � @ � @ � @ � @ � @ @ � @ � @ � � @ � @ @ � @ � @ � � @ � @ @ � @ � @ � @ � @ � @ � @ � @ � @ � � @ � @ � @ � @ � @ � @ � � � � � � � � � � � � � � � � � � � � | {z } � D E = 196 GP a � = 0 : 3 g = 0 2 ; 1 8 F = 1000 N on the upp er pa rt > < of the b ounda ry g = 2 ; 2 > : 0 otherwise � � Examples 2.5 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Classical Galerkin App roach � Select a �nite dimensional subspace U � U . h � T o solve min E ( u ) h u 2U h h � Evaluate of E ( v ) fo r v 2 U , e.g. h h h Z 1 0 0 E ( v ) = d ( x ) v v � v f dx h h h h 0 � Select a �nite element basis fo r U (small sup- p o rt) 0 � F o rm derivatives v fo r the basis functions an- h alytically � Compute integrals numerically � Assemble sti�ness matrix � Solve (linea r) system. � � Numerical T echniques 3.1 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Direct Quadrature App roach � No dal values (on mesh) u = ( u ) ; u = u ( x ) i i =0 ::: n i i h � Find di�erentiation/quadrature rules fo r E ( u ) � E ( u ) directly . h h � Solve min E ( u ) h h n u 2 R h Example Z 1 0 0 E ( u ) = u ( x ) u ( x ) + u ( x ) f ( x ) dx 0 � � n 2 X u � u u f + u f i i � 1 i i i � 1 i � 1 E ( u ) = h + h h h 2 i =1 � u + 2 u � u i � 1 i i � 1 No rmal equations: = f i 2 h � � Numerical T echniques 3.2 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Direct Quadrature App roach Problem Minimum do es not exist fo r n= 2 � 1 � � 2 u � u X 2 j 2 j +2 E ( u ) = 2 h + u f h h 2 j +1 2 j +1 2 h j =0 � � Numerical T echniques 3.3 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Basic theo ry fo r linea r p roblem 2 Theo rem: Given Hilb ert space V , jj � jj = < � ; � > � bilinea r fo rms a ( � ; � ), ~ a ( � ; � ), f 2 V 2 2 c jj v jj � a ( v ; v ) � c jj v jj 8 v 2 V , W � V 1 2 ~ a ( v ; v ) � a ( v ; v ) 8 v 2 V 2 j ~ a ( v ; v ) � a ( v ; v ) j � � jj v jj 8 v 2 W min E ( u ) = min a ( u; u ) � 2 f ( u ) u 2 V u 2 V ~ min E ( u ) ~ = min ~ a ( u; ~ u ~ ) � 2 f ( u ) ~ u 2 V ~ u 2 V ~ min E ( w ) = min a ( w ; w ) � 2 f ( w ) w 2 W w 2 W Then � � 2 2 2 jj u � ~ u jj � c � jj w jj + jj w � u jj 0 Application: W solution (FE) space, V auxilia ry la rger (FE) space fo r de�ning integration, a co r- rect, ~ a numerical bilinea r fo rm. � � Numerical T echniques 3.4 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Asymptotic expansions Di�erentiati on b y basic di�erences and integra- tion b y simple mid p oint rule (and their 2D gen- eralization) lead to even asymptotic expansions of the fo rm 2 4 E ( u ) = E ( u ) + h E ( u ) + h E ( u ) � � � 2 4 h Reference: 1D case: Lyness 1968; 2D case: R. 1993, R. and Lyness 1994. Extrap olation of functionals � � 4 1 min E ( u ) � E ( u ) h h 2 h h 3 3 � � 64 20 1 min E ( u ) � E ( u ) + E ( u ) h h 2 h h 4 h h 45 45 45 � � � � � Numerical T echniques 3.5 � �

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Application to Example 2 ! Z � � 1 4 2 u 0 2 E ( u ) = u ( x ) � c � u dx: 2 0 h � 1 = 4 ; 1 = 8 ; 1 = 16 ; 1 = 32 ; 1 = 64 Order 2 Order 4 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 � � Numerical T echniques 3.6 � � 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Matrix structure (o rder 2,4,6,8) 0 0 10 10 20 20 30 30 40 40 50 50 60 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 nz = 187 nz = 247 0 0 10 10 � � Numerical T echniques 3.7 � � 20 20 30 30 40 40 50 50 60 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 nz = 275 nz = 287

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Erro r fo r example 2 Metho ds of (fo rmal) o rder 2,4,6 2 L -Erro r versus numb er of unkno wns 0 10 Erro r fo r 6th o rder -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 0 1 2 3 10 10 10 10 � � Numerical T echniques 3.7 -6 � 2 x 10 � 0 -2 -4 -6 -8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at � f� ur Mathematik r r r r r r r r r r Ulrich R� ude ' $ Hiera rchical Basis L;H Sti�ness matrix in hiera rchical basis K l ~ sti�ness matrix fo r coa rse mesh no des only K l � 1 L;H ~ fo r right hand sides f and f , resp ectively . l l � 1 0 1 L;N 4 L;H K K l � 1 l ;v m 4 L;H 1 B 3 C Original Function ~ K � K = @ A l l � 1 4 L;H 4 L;H 3 3 Hierarchical Function K K l ;mv l ;mm 3 3 Hierarchical Displacements � � Hiera rchical Basis and Multigrid 4.1 � �