Isogeometric Analysis for high-order two-point singularly perturbed - PowerPoint PPT Presentation

Isogeometric Analysis for high-order two-point singularly perturbed problems of reaction-diffusion type Christos Xenophontos Department of Mathematics and Statistics University of Cyprus

Outline ► The model problem  Regularity assumptions ► IGA Galerkin formulation ► Error Analysis ► Numerical Examples ► Closing remarks 1

The Model Problem Following [ Sun and Stynes , 1995] , let ν  1 be an integer and consider following problem: find u  C 2 ν ( I ), Ι = (0, 1) such hat ( )   − ( 1)  −    + −  −  − + =  2 (2 ) 1 ( 1) ( 1) ( 1) in L u u a u L u f I   − 2( 1) 1   = = =  − ( ) ( ) j j  (0) (1) 0 , 0,1,..., 1 u u j 2

The Model Problem Following [ Sun and Stynes , 1995] , let ν  1 be an integer and consider following problem: find u  C 2 ν ( I ), Ι = (0, 1) such hat ( )   − ( 1)  −    + −  −  − + =  2 (2 ) 1 ( 1) ( 1) ( 1) in L u u a u L u f I   − 2( 1) 1   = = =  − ( ) ( ) j j  (0) (1) 0 , 0,1,..., 1 u u j where ε  (0, 1] is a perturbation parameter and  =  0 , 1     ( ) L u  −   ( ) k  −  − +  − − +   1 ( 1) ( ) k k k ( 1) , 1 a u a u   − +  − 2( ) 1 2( ) k k = 2 k 2

Examples : −   + =  2 ( ) ( ) ( ) , (0,1), u x u x f x x ν = 1  = =  (0) (1) 0 u u 3

Examples : −   + =  2 ( ) ( ) ( ) , (0,1), u x u x f x x ν = 1  = =  (0) (1) 0 u u    − + =  2 (4) ( ) ( ) ( ) ( ) , (0,1), u x u x u x f x x ν = 2    = = = =  (0) (1) (0) (1) 0 u u u u 3

Examples : −   + =  2 ( ) ( ) ( ) , (0,1), u x u x f x x ν = 1  = =  (0) (1) 0 u u    − + =  2 (4) ( ) ( ) ( ) ( ) , (0,1), u x u x u x f x x ν = 2    = = = =  (0) (1) (0) (1) 0 u u u u −   + − + =  2 (6) (4) ( ) ( ) ( ) ( ) ( ) , (0,1), u x u x u x u x f x x ν = 3      = = = = = =  (0) (1) (0) (1) (0) (1) 0 u u u u u u etc. 3

     We assume that and 1) ( ) 0 [0,1] a x x  − 2( 1  −      =  ( ) ( ) 0 [0,1], 2,..., a x a x x k  −  − +  − 2( k ) ( k ) 1 k 2 for some constants α ν – k , k = 1, …, ν, ( α ν – 1 = α ), such that k    =  0 , 2,..., k  − j = 1 j 4

     We assume that and 1) ( ) 0 [0,1] a x x  − 2( 1  −      =  ( ) ( ) 0 [0,1], 2,..., a x a x x k  −  − +  − 2( k ) ( k ) 1 k 2 for some constants α ν – k , k = 1, …, ν, ( α ν – 1 = α ), such that k    =  0 , 2,..., k  − j = 1 j The above conditions ensure coercivity of the associated bilinear form (as well as the absence of turning points) [ Sun and Stynes , 1995] . 4

We further assume that the data is analytic , i.e. the functions a i , i = 0, 1, …, 2( ν – 1), and the function f satisfy, for some positive constants C , γ f , γ a i , i = 0, 1, …, 2( ν – 1) independent of ε ,       ( ) ( ) n n n n ! , ! a C n f C n n   0 i a f ( ) ( ) L I i L I 5

Assumption 1 : The BVP under study has a classical solution ( )   2 which can be decomposed as u C I  = + + u u u u S BL R 6

Assumption 1 : The BVP under study has a classical solution ( )   2 which can be decomposed as u C I  = + + u u u u S BL R smooth part (as smooth as the data allows) 6

Assumption 1 : The BVP under study has a classical solution ( )   2 which can be decomposed as u C I  = + + u u u u S BL R smooth part (as smooth as the data allows) boundary layers (have support only in a region near the boundary) 6

Assumption 1 : The BVP under study has a classical solution ( )   2 which can be decomposed as u C I  = + + u u u u S BL R remainder (exponentially small in ε ) smooth part (as smooth as the data allows) boundary layers (have support only in a region near the boundary) 6

Assumption 1 : The BVP under study has a classical solution ( )   2 which can be decomposed as u C I  = + + u u u u S BL R remainder (exponentially small in ε ) smooth part (as smooth as the data allows) boundary layers (have support only in a region near the boundary) and for all x  [0, 1], n  , there holds (under the analyticity of the data assumption) 6

  ( ) n n !, u C n  S S ( ) L I ( ) ( ) n      − − −   1 dist( , )/ n x I ( ) , u x C e BL BL  −    + +  ( ) / u u u Ce  −   1 R R R 2 ( ) ( ) H I L I ( ) L I      for some constants , independent of ε . , , , 0 S BL 7

  ( ) n n !, u C n  S S ( ) L I ( ) ( ) n      − − −   1 dist( , )/ n x I ( ) , u x C e BL BL  −    + +  ( ) / u u u Ce  −   1 R R R 2 ( ) ( ) H I L I ( ) L I      for some constants , independent of ε . , , , 0 S BL Moreover, there exist C , K > 0, such that for all n = 1, 2, 3, …    − −   ( ) 1 n n n n max , u CK n  ( ) L I 7

  ( ) n n !, u C n Differentiability  S S ( ) L I ( )  through asymptotic ( ) n      − − −   1 dist( , )/ n x I ( ) , u x C e BL BL expansions  −    + +  ( ) / u u u Ce  −   1 R R R 2 ( ) ( ) H I L I ( ) L I      for some constants , independent of ε . , , , 0 S BL Moreover, there exist C , K > 0, such that for all n = 1, 2, 3, …    − −   ( ) 1 n n n n max , u CK n  ( ) L I  7 Classical Differentiability

Remark: Assumption 1 has been established for ν = 1, [ Melenk , 1997], and for ν = 2, [ Constantinou , 2019] . 8

IGA Galerkin Formulation   0 ( ) The variational formulation is given by: find u H I such that =    B ( , ) , ( ) u v f v v H I  0 I   where, with the usual L 2 ( I ) inner product, , I =    +  −  − + B B 2 ( ) ( ) ( 1) ( 1) ( , ) , , ( , ) v w v w a v w v w   − 2( 1) 1 I I 9

IGA Galerkin Formulation   0 ( ) The variational formulation is given by: find u H I such that =    B ( , ) , ( ) u v f v v H I  0 I   where, with the usual L 2 ( I ) inner product, , I =    +  −  − + B B 2 ( ) ( ) ( 1) ( 1) ( , ) , , ( , ) v w v w a v w v w   − 2( 1) 1 I I  =  0 , 1    B  ( , ) v w    − +  −  − +   1 ( 1) ( ) ( ) k k k , , 1 a v a v w   − +  − 2( ) 1 2( ) k k = I 2 k 9

   At the discrete level, we seek ( ) s. t. u S H I 0 N N =     B ( , ) , ( ) u v f v v S H I  0 N N I and there holds + −  −    , u u C u v v S C N N E E 10

   At the discrete level, we seek ( ) s. t. u S H I 0 N N =     B ( , ) , ( ) u v f v v S H I  0 N N I and there holds + −  −    , u u C u v v S C N N E E where the energy norm is defined as 2 2  2 =  + 2 ( ) w w w  − 1 2 ( ) E H I ( ) L I 10

In the FEM, the space S N consists of piecewise polynomials defined on some subdivision (mesh) of the domain Ω . 11

In the FEM, the space S N consists of piecewise polynomials defined on some subdivision (mesh) of the domain Ω . In IGA, the space S N is defined using B -splines:   N  =  , ( ) S S span B  , N p i p = 1 i 11

In the FEM, the space S N consists of piecewise polynomials defined on some subdivision (mesh) of the domain Ω . In IGA, the space S N is defined using B -splines:   N  =  , ( ) S S span B  , N p i p = 1 i The functions B i,p are B -splines, defined as follows [ Cotrell, Hughes, Basilevs, 2009] : 11

Let    =    , ,..., + + 1 2 1 N p   be a knot vector , where is the i th knot, i = 1,…, N + p +1, i p is the polynomial order and N is the total number of basis functions. 12