A Priori Error Analysis of Fully Discrete Elliptic model problem - PowerPoint PPT Presentation

A Priori Error Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for Elliptic Problems A Priori Error Analysis of Fully Discrete Elliptic model problem First convergence results FE-HMM Error Analysis of the Fully Discrete FE-HMM Fully discrete FE problem Monika Wolfmayr Convergence results for the macrosolution H 1-error L 2-error L 2-projection of u ε Convergence results 5th December 2011 for the fully discrete solution H 1-error Conclusions

A Priori Error Outline Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction Introduction FE-HMM for FE-HMM for Elliptic Problems Elliptic Problems Elliptic model problem Elliptic model problem First convergence results First convergence results Error Analysis of the Fully Discrete FE-HMM Error Analysis of the Fully Discrete FE-HMM Fully discrete FE problem Fully discrete FE problem Convergence results for the macrosolution H 1-error Convergence results for the macrosolution L 2-error H 1 -error L 2-projection of u ε L 2 -error Convergence results for the fully discrete solution L 2 -projection of u ε H 1-error Convergence results for the fully discrete solution Conclusions H 1 -error Conclusions

A Priori Error Introduction Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction Heterogeneous Multiscale Method (HMM) introduced by E FE-HMM for and Engquist in 2003 Elliptic Problems Elliptic model problem First convergence results the name heterogeneous was used to emphasize that the Error Analysis of models at different scales may be of very different nature the Fully Discrete FE-HMM Fully discrete FE problem Difference between traditional MM and HMM: Convergence results for the macrosolution H 1-error L 2-error MM: general purpose are microscale solvers, i.e. to resolve L 2-projection of u ε the details of the solutions of the microscale model Convergence results for the fully discrete solution H 1-error HMM: objective is to capture the macroscale behavior of the Conclusions system with a cost that is much less than the cost of full microscale solvers

A Priori Error Introduction Analysis of Fully Discrete FE-HMM HMM: Monika Wolfmayr physical problem is directly discretized by a macroscopic Introduction finite element method model (coarse scale) FE-HMM for microproblems are either unit-cell problems or problems on a Elliptic Problems Elliptic model problem patch with a fixed number of unit cells (fine scale) First convergence results Error Analysis of study of accuracy properties in HMM: the Fully Discrete FE-HMM ◮ first approach: assumption that the microproblems are Fully discrete FE problem analytically given; macro- and microerrors often Convergence results for the macrosolution H 1-error separately estimated L 2-error L 2-projection of u ε ◮ further approach: combination of microscopic and Convergence results for the fully discrete macroscopic models; microproblems are solved solution H 1-error numerically as well; estimates for the errors transmitted Conclusions on the macroscale by discretizing the fine scale (Abdulle, 2005) analysis for piecewise linear continuous FEMs in the micro- and macrospaces and for the periodic case

A Priori Error Notation Analysis of Fully Discrete FE-HMM Monika Wolfmayr r = ( r 1 , ..., r n ) ∈ N n , | r | = r 1 + ... + r n , D r = ∂ r 1 1 ...∂ r n n ; Introduction FE-HMM for Elliptic Problems H 1 (Ω) = { u ∈ L 2 (Ω); D r u ∈ L 2 (Ω) , | r | ≤ 1 } , Elliptic model problem First convergence results   1 / 2 Error Analysis of  � � D r u � 2 the Fully Discrete � u � H 1 (Ω) = ;  L 2 (Ω) FE-HMM Fully discrete FE | r |≤ 1 problem Convergence results for the macrosolution H 1-error L 2-error W l , ∞ (Ω) = { u ∈ L ∞ (Ω); D r u ∈ L ∞ (Ω) , | r | ≤ l } ; L 2-projection of u ε Convergence results � for the fully discrete solution W 1 per ( Y ) = { v ∈ H 1 per ( Y ); v dx = 0 } ; H 1-error Y Conclusions �·� H 1 , H 1 �·� H 1 , Y = (0 , 1) n H 1 0 (Ω) = C ∞ per ( Y ) = C ∞ 0 (Ω) per ( Y )

A Priori Error Elliptic Model Problem Analysis of Fully Discrete FE-HMM Monika Wolfmayr −∇ · ( a ε ∇ u ε ) = f in Ω ⊂ R n Introduction (1) u ε = 0 on ∂ Ω , FE-HMM for Elliptic Problems Elliptic model problem First convergence ε ... length scale, Ω ... convex polygon, f ∈ L 2 (Ω), results Error Analysis of a ε ( x ) = a ( x , x ε ) = a ( x , y ) ... symmetric and coercive tensor, the Fully Discrete FE-HMM periodic with respect to each component in Y = (0 , 1) n , Fully discrete FE problem a ij ( x , · ) ∈ L ∞ ( R n ) Convergence results for the macrosolution H 1-error L 2-error u ε converges weakly to a homogenized solution u 0 of L 2-projection of u ε Convergence results for the fully discrete −∇ · ( a 0 ∇ u 0 ) = f solution in Ω H 1-error (2) u 0 = 0 Conclusions on ∂ Ω , a 0 ... smooth matrix with coefficients � k =1 a ik ( x , y ) ∂χ j Y ( a ij ( x , y ) + � n a 0 ij ( x ) = ∂ y k ( x , y )) dy , χ j ( x , · ) ... solution of the cell problems

A Priori Error Elliptic Model Problem Analysis of Fully Discrete FE-HMM Macro FE space: Monika Wolfmayr 0 (Ω , T H ) = { u H ∈ H 1 S 1 0 (Ω); u H | K ∈ P 1 ( K ) ∀ K ∈ T H } , (3) Introduction FE-HMM for P 1 ( K ) ... space of linear polynomials on the triangle K , Elliptic Problems Elliptic model problem T H ... quasi-uniform triangulation of Ω of shape regular K , First convergence results H ... size of triangulation Error Analysis of the Fully Discrete FE-HMM Macrobilinear form: Fully discrete FE problem � Convergence results | K | � for the macrosolution B ( u H , v H ) = ∇ u a ( x k , x /ε )( ∇ v ) T dx , (4) H 1-error | K ε | L 2-error K ε L 2-projection of u ε K ∈T H Convergence results K ε = x k + ε [ − 1 / 2 , 1 / 2] n ... sampling subdomain for the fully discrete solution H 1-error Conclusions u is the solution of the exact microproblem : Find u such that ( u − u H ) ∈ W 1 per ( K ε ) and � ∀ z ∈ W 1 ∇ u a ( x k , x /ε )( ∇ z ) T dx = 0 per ( K ε ) . (5) K ε

A Priori Error Elliptic Model Problem Analysis of Fully Discrete FE-HMM Monika Wolfmayr Introduction FE-HMM for It can be shown that Elliptic Problems Elliptic model problem n First convergence χ j ( x k , x /ε ) ∂ u H ( x k ) � u = u H + ε results , (6) Error Analysis of ∂ x j the Fully Discrete j =1 FE-HMM Fully discrete FE problem χ j ( x k , y ) ... unique solutions of the cell problems: Convergence results for the macrosolution H 1-error � � L 2-error ∇ χ j a ( x k , y )( ∇ z ) T dy = − e T j a ( x k , y )( ∇ z ) T dy L 2-projection of u ε (7) Convergence results for the fully discrete Y Y solution H 1-error for all z ∈ W 1 per ( Y ); { e j } n j =1 ... standard basis in R n . Conclusions

A Priori Error Elliptic Model Problem Analysis of Fully Discrete FE-HMM Monika Wolfmayr Variational problem for the macrosolution: Introduction Find u H ∈ S 1 0 (Ω , T H ) such that FE-HMM for Elliptic Problems ∀ v H ∈ S 1 Elliptic model problem B ( u H , v H ) = � f , v H � 0 (Ω , T H ) . (8) First convergence results Error Analysis of B ( · , · ) elliptic, bounded ⇒ unique solution of (8) the Fully Discrete FE-HMM Fully discrete FE problem It can be shown that Convergence results for the macrosolution H 1-error � L 2-error � ∇ u H a 0 ( x k )( ∇ v H ) T dx . L 2-projection of u ε B ( u H , v H ) = (9) Convergence results for the fully discrete K K ∈T H solution H 1-error Conclusions Remark: Since we assume an exact microsolver, so far the variational problem for the macrosolution (8) is of semidiscrete nature.

A Priori Error First convergence estimates for the macrospace Analysis of Fully Discrete FE-HMM Monika Wolfmayr Assumption: H 2 -regularity for u 0 , exact microsolution Introduction � u ε − u 0 � L 2 (Ω) ≤ C ε � f � L 2 (Ω) , FE-HMM for (10) Elliptic Problems Elliptic model problem � u ε − u H � H 1 (Ω) ≤ C ( H /ε ) � f � L 2 (Ω) , First convergence (11) results � u 0 − u H � H 1 (Ω) ≤ C H � f � L 2 (Ω) , Error Analysis of (12) the Fully Discrete H 1 (Ω) ≤ C ( √ ε + H ) � f � L 2 (Ω) , FE-HMM � u ε − u ε p � ¯ (13) Fully discrete FE problem Convergence results for the macrosolution p ... reconstructed solution from u H with ( u − u H ) H 1-error u ε L 2-error L 2-projection of u ε periodically extended on each K , can be discontinuous Convergence results for the fully discrete across K , hence ¯ H 1 -norm is mesh-dependent; solution H 1-error Conclusions � Pu ε − u H � H 1 (Ω) ≤ C ( ε/ H + H ) � f � L 2 (Ω) , (14) Pu ε ... L 2 -projection of the solution