On the Gradgrad and divdiv Complexes, and a Related Decomposition Result for Biharmonic Problems in 3D: Part 2

Transcription

1 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 On the Gradgrad and dvdv Complexes, and a Related Decomposton Result for Bharmonc Problems n 3D: Part 2 Drk Pauly (Fakultät für Mathematk, UDE, Germany) & Walter Zulehner (Insttut für Numersche Mathematk, JKU, Austra) RICAM Specal Semester 216 Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems October 17-21, 216, Lnz, Austra

2 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Problem Kernel of dv Dv Walter s central queston N(dv Dv) =? more precsely: kernel N(dv Dv S ) =? of dv Dv S D(dv Dv S ) L 2 (Ω, S) L 2 (Ω, R), wth Ω (open) R 3 and M dv Dv M D(dv Dv S ) = {M L 2 (Ω, S) dv Dv L 2 (Ω, R)} note n general wth Dv M H 1 (Ω, R 3 ) dv Dv S (dv)(dv S ) D(dv)(Dv S ) = {M L 2 (Ω, S) Dv M L 2 (Ω, R 3 ) and dv Dv M L 2 (Ω, R)}

3 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 De-Rahm Complexes grad-rot-dv Complexes sequence or complex D( grad) grad D( rot) wth dual or adjont sequence or complex L 2 (Ω) dv D(dv) rot D( dv) rot D(rot) dv grad D(grad) unbounded, densely defned, closed, lnear operators wth adjonts spaces grad D( grad) L 2 L 2, ( grad) = dv D(dv) L 2 L 2, rot D( rot) L 2 L 2, dv D( dv) L 2 L 2, ( rot) = rot D(rot) L 2 L 2, ( dv) = grad D(grad) L 2 L 2 π R R ι R R D( grad) = H 1 = H 1 = H (grad), D( rot) = R = H (rot) = H (curl), D( dv) = D = H (dv), D(dv) = D = H(dv), D(rot) = R = H(rot) = H(curl), D(grad) = H 1 = H 1 = H(grad)

4 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 De-Rahm Complexes grad-rot-dv Complexes sequence or complex D( grad) grad D( rot) wth dual or adjont sequence or complex L 2 (Ω) dv D(dv) rot D( dv) rot D(rot) complex property: range kernel (rot grad =, dv rot = ) R() = {} = N( grad), R( grad) N( rot), dv grad D(grad) R( rot) N( dv), π R R ι R R R( dv) = N(π R ) = R, R(ι R ) = R = N(grad), R( grad) N(rot), R(rot) N( dv), R( dv) = N() = L 2 (Ω) complex closed all ranges are closed complex exact all cohomology groups are trval,.e., f H D = N( rot) R( grad) = N(dv) R(rot) = N( rot) N(dv) = {} H N = N( dv) R( rot) = N(rot) R(grad) = N( dv) N(rot) = {} (Drchlet felds), (Neumann felds) dmenson depends only on topology, Bett numbers

5 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 De-Rahm Complexes grad-rot-dv Complexes sequence or complex: D( grad) grad D( rot) wth dual or adjont sequence or complex L 2 (Ω) dv D(dv) rot D( dv) rot D(rot) dv grad D(grad) crucal : compact embeddngs (for, e.g., Ω bounded and strong Lpschtz) D(grad) L 2 (Rellch s selecton theorem), D( rot) D(dv) L 2 (Weck s selecton theorem, Weck 74), D(rot) D( dv) L 2 (Weck s selecton theorem) closed complexes and fnte cohomology groups Helmholtz decompostons Fredrchs/Poncaré type estmates contnuous and compact nverses of reduced operators...all from general fa-toolbox π R R ι R R

6 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox General Complexes settng: unbounded, densely defned, closed, lnear operators wth adjonts A D(A ) H H +1, A D(A ) H +1 H, Z sequence or complex wth adjont:... D(A 1 )... A 1 D(A ) A 1 D(A 1 ) A D(A +1 ) complex: range kernel,.e., A A 1 =, A 1 A A A D(A ) A +1 D(A +1 )... =,.e., R(A 1 ) N(A ), R(A ) N(A 1 ) related problem: fnd x D(A ) D(A 1 ) s.t. A x = f, A 1 x = g, π x = h, where f R(A ), g R(A 1 ) and h H wth kernel/cohomology group H = N(A ) N(A 1 )

7 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Reduced Operators and Compact Embeddngs Hodge/Helmholtz/Weyl decompostons: H = N(A ) H R(A ), H +1 = N(A ) H +1 R(A ) reduced (njectve) operators A D(A ) = D(A ) R(A ) R(A ) R(A ), A D(A ) H H +1 A D(A ) = D(A ) R(A ) R(A ) R(A ), A D(A ) H +1 H A 1, (A ) 1 exst, exact sequence for A, A crucal: compact embeddngs note D(A ) H D(A ) H +1 D(A 1 ) H 1 D(A ) H H H D(A ) D(A 1 ) H (general) Poncaré type estmates (Poncaré, Fredrchs, Maxwell,...) closed ranges contnuous and compact nvers operators Helmholtz decompostons

8 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Fredrchs/Poncaré Type Estmates compact embeddng D(A ) H ϕ D(A ) ϕ H c A A ϕ H+1 ψ D(A ) ψ H +1 c A A ψ H R(A ) = R(A ), R(A ) = R(A ) closed reduced operators A D(A ) = D(A ) R(A ) R(A ) R(A ), A D(A ) H H +1 A D(A ) = D(A ) R(A ) R(A ) R(A ), A D(A ) H +1 H A 1 R(A ) D(A ) cont., A 1 R(A ) R(A ) cpt., A 1 = c A (A ) 1 R(A ) D(A ) cont., (A ) 1 R(A ) R(A ) cpt., (A ) 1 = c A note: best constants c A and c A satsfy 1 A ϕ H+1 A = nf = nf ψ H = 1 c = c A = c c A ϕ D(A ) ϕ H ψ D(A ) ψ H+1 c A A

9 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Fredrchs/Poncaré Type Constants recall: compact embeddng D(A ) H Lemma ϕ D(A ) ψ D(A ) c = c A = c A ϕ H c A A ϕ H+1,wthbestcons. ψ H +1 c A A ψ H,wthbestconst. 1 A ϕ H+1 = nf c A ϕ D(A ) ϕ H 1 A = nf ψ H c A ψ D(A ) ψ H+1 Proof. ϕ D(A ) = D(A ) R(A ), R(A ) = R(A ) closed ϕ = A ψ wth ψ D(A ) ϕ 2 H = ϕ, A ψ H = A ϕ, ψ H+1 A ϕ H+1 ψ H+1 c A A ϕ H+1 A ψ =ϕ symmetry ϕ H c A A ϕ H+1 c A c A c A c A H

10 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Helmholtz Type Decomposton H = N(A ) H R(A ), H = R(A 1 ) H N(A 1 ) D(A ) = N(A ) H D(A ), D(A 1 ) = D(A 1 ) H N(A 1 ) exact sequence: R(A 1 ) N(A ), R(A ) N(A 1 ) N(A 1 ) = H H R(A ), N(A ) = R(A 1 ) H H, H = N(A ) N(A 1 ) refned Helmholtz decomposton H = R(A 1 ) H H H R(A ) D(A ) = R(A 1 ) H H H D(A ) D(A 1 ) = D(A 1 ) H H H R(A ) wth orthonormal projectors π A 1 H R(A 1 ), ψ D(A 1 ) π A 1 ψ D(A 1 ) A 1 π A 1 ψ = A 1 ψ π A H R(A ), ϕ D(A ) π A ϕ D(A ) A π A ϕ = A ϕ π H H

11 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Soluton Theory problem: fnd x D(A ) D(A 1 ) s.t. A x = f A 1 x = g π x = h Theorem (soluton theory) unque soluton (dpd. cont. on data) f R(A ), g R(A 1 ) and h H Proof. x = A 1 f + (A 1 ) 1 g + h

12 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Varatons (Saddle Pont) Formulatons unque soluton x = A 1 f + (A 1 ) 1 g + h D(A ) D(A 1 ) of A x = f, A 1 x = g, π x = h can be found by varatonal technques (Lax-Mlgram) for A 1 f we solve A A ψ = f :fndψ D(A ) wth ϕ D(A ) A ψ,a ϕ H = f,ϕ H+1 (1) f R(A ) (1) holds for all ϕ D(A ) x A = A ψ D(A ) and A x A = f x A = A 1 f D(A ) and x A H c f H+1 note: D(A ) = D(A ) R(A ) and R(A ) = N(A ) H +1 (1) s equvalent to the saddle pont problem: fnd ψ D(A ) wth ϕ D(A ) A ψ,a ϕ H = f,ϕ H+1, φ N(A ) = R(A +1 ) H +1 H +1 ψ,φ H+1 =

13 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Varatons (Saddle Pont) Formulatons unque soluton x = A 1 f + (A 1 ) 1 g + h D(A ) D(A 1 ) of A x = f, A 1 x = g, π x = h can be found by varatonal technques (Lax-Mlgram) for (A 1 ) 1 g we solve A 1 A 1ψ = f :fndψ D(A 1 ) wth ϕ D(A 1 ) A 1 ψ,a 1 ϕ H = g,ϕ H 1 (2) g R(A 1 ) (2) holds for all ϕ D(A 1) x A 1 = A 1 ψ D(A 1 ) and A 1 x A 1 = g x A 1 = (A 1 ) 1 g D(A 1 ) and x A 1 H c 1 g H 1 note: D(A 1 ) = D(A 1 ) R(A 1 ) and R(A 1 ) = N(A 1) H 1 (2) s equvalent to the saddle pont problem: fnd ψ D(A 1 ) wth ϕ D(A 1 ) A 1 ψ,a 1 ϕ H = g,ϕ H 1, φ N(A 1 ) = R(A 2 ) H H 1 ψ,φ H 1 =

14 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Grad grad and dv Dv Complexes complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) wth dual or adjont complex dv Dv S D(dv Dv S ) Rot S D( Dv T ) sym Rot T D(sym Rot T ) unbounded, densely defned, closed, lnear operators wth adjonts Dv T dev Grad D(dev Grad) π RT RT ι RT RT Grad grad D( Grad grad) L 2 R L2 S, ( Grad grad) = dv Dv S D(dv Dv S ) L 2 S L2 R, Rot S D( Rot S ) L 2 S L2 T, Dv T D( Dv T ) L 2 T L2 R 3, settng n natural Hlbert spaces one can show D( Grad grad) = H 2, D(dev Grad) = H 1 ( Rot S ) = sym Rot T D(sym Rot T ) L 2 T L2 S, ( Dv T ) = dev Grad D(dev Grad) L 2 R 3 L 2 T

15 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Results for Grad grad and dv Dv Complexes (trv. top.) complex and adjont complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) dv Dv S D(dv Dv S ) general assumpton: Rot S D( Dv T ) sym Rot T D(sym Rot T ) Dv T dev Grad D(dev Grad) Ω R 3 (R n ) bounded strong Lpschtz doman π RT RT ι RT RT Lemma Let Ω be addtonally topologcally trval. Then: N( Rot S ) = R( Grad grad) = Grad grad H 2 N( Dv T ) = R( Rot S ) = Rot H 1 S RT L 2 = R( Dv T ) = Dv H 1 T N(sym Rot T ) = R(dev Grad) = dev Grad H 1 N(dv Dv S ) = R(sym Rot T ) = sym Rot H 1 T (orgnal queston) L 2 = R(dv Dv S ) = dv Dv H 2 S Especally, all ranges are closed and admt regular potentals. both complexes are closed

16 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Results for Grad grad and dv Dv Complexes (trv. top.) complex and adjont complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) dv Dv S D(dv Dv S ) general assumpton: Lemma Rot S D( Dv T ) sym Rot T D(sym Rot T ) Dv T dev Grad D(dev Grad) Ω R 3 (R n ) bounded strong Lpschtz doman π RT RT ι RT RT Let Ω be addtonally topologcally trval. Then all cohomology groups are trval,.e., H D S = N( Rot S ) N(dv Dv S ) = N( Rot S ) R( Grad grad) = N(dv Dv S ) R(sym Rot T ) = {} H N T = N( Dv T ) N(sym Rot T ) = N( Dv T ) R( Rot S ) = N(sym Rot T ) R(dev Grad) = {} (The symmetrc Drchlet felds and trace-free Neumann felds vansh.) Moreover, the followng Helmholtz type decompostons hold: L 2 (Ω, S) = N( Rot S ) L 2 (Ω,S) N(dv Dv S), L 2 = R(dv Dv S ) N( Rot S ) = R( Grad grad), N(dv Dv S ) = R(sym Rot T ), L 2 (Ω, T) = N( Dv T ) L 2 (Ω,T) N(sym Rot T), RT L 2 = R( Dv T ), N( Dv T ) = R( Rot S ), both complexes are closed and exact N(sym Rot T ) = R(dev Grad).

17 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Results for Grad grad and dv Dv Complexes (gen. top.) complex and adjont complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) dv Dv S D(dv Dv S ) Rot S D( Dv T ) sym Rot T D(sym Rot T ) Dv T dev Grad D(dev Grad) π RT RT ι RT RT general assumpton: Ω R 3 (R n ) bounded strong Lpschtz doman Lemma The followng embeddngs are compact: D( Rot S ) D(dv Dv S ) L 2 (Ω, S), D( Dv T ) D(sym Rot T ) L 2 (Ω, T) proof: partton of unty, results for trv. top. (HD and reg. pot.) and Lemma It holds drectly and topologcally {M L 2 (Ω, S) dv Dv M H 1 (Ω)} = H 1 Id + N(dv Dv S ).

18 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Results for Grad grad and dv Dv Complexes (gen. top.) complex and adjont complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) dv Dv S D(dv Dv S ) Rot S D( Dv T ) sym Rot T D(sym Rot T ) Dv T dev Grad D(dev Grad) π RT RT ι RT RT general assumpton: Ω R 3 (R n ) bounded strong Lpschtz doman Lemma The cohomology groups H D S, HN T (symmetrc Drchlet felds and trace-free Neumann felds) are fnte dmensonal. Moreover, the followng Helmholtz type decompostons hold: L 2 (Ω, S) = R( Grad grad) L 2 (Ω,S) HD S L 2 (Ω,S) R(sym Rot T), N( Rot S ) = R( Grad grad) L 2 (Ω,S) HD S, N(dv Dv S) = R(sym Rot T ) L 2 (Ω,S) HD S, L 2 (Ω, T) = R( Rot S ) L 2 (Ω,T) HN T L 2 (Ω,T) R(dev Grad), N( Dv T ) = R( Rot S ) L 2 (Ω,T) HN T, N(sym Rot T) = R(dev Grad) L 2 (Ω,T) HN T. Especally, all ranges are closed. both complexes are closed

19 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Results for Grad grad and dv Dv Complexes (gen. top.) complex and adjont complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) dv Dv S D(dv Dv S ) Rot S D( Dv T ) sym Rot T D(sym Rot T ) Dv T dev Grad D(dev Grad) π RT RT ι RT RT general assumpton: Ω R 3 (R n ) bounded strong Lpschtz doman Lemma The followng Fredrchs/Poncaré type estmates hold: c Gg, c D, c R > s.t. u H 2 u L 2 (Ω) c Gg Grad grad u L 2 (Ω), M D(dv Dv S ) R( Grad grad) M L 2 (Ω) c Gg dv Dv M L 2 (Ω), N D( Dv T ) R(dev Grad) N L 2 (Ω) c D Dv N L 2 (Ω), E H 1 RT L 2 E L 2 (Ω) c D dev Grad E L 2 (Ω), M D( Rot S ) R(sym Rot T ) M L 2 (Ω) c R Rot M L 2 (Ω), N D(sym Rot T ) R( Rot S ) N L 2 (Ω) c R sym Rot N L 2 (Ω) note: same constants!

20 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Results for Grad grad and dv Dv Complexes (gen. top.) complex and adjont complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) dv Dv S D(dv Dv S ) Rot S D( Dv T ) sym Rot T D(sym Rot T ) Dv T dev Grad D(dev Grad) π RT RT ι RT RT general assumpton: Ω R 3 (R n ) bounded strong Lpschtz doman Lemma The respectve reduced operators have contnuous resp. compact nverses, e.g., ( Rot S ) 1 R( Rot S ) cont. D( Rot S ) R(sym Rot T ) cpt. R(sym Rot T ), (sym Rot T ) 1 R(sym Rot T ) cont. D(sym Rot T ) R( Rot S ) cpt. R( Rot S ),

21 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Grad grad and dv Dv Complexes Results for Grad grad and dv Dv Complexes (trv. top.) complex and adjont complex D( Grad grad) L 2 (Ω) Grad grad D( Rot S ) dv Dv S D(dv Dv S ) Rot S D( Dv T ) sym Rot T D(sym Rot T ) Dv T dev Grad D(dev Grad) π RT RT ι RT RT general assumpton: Ω R 3 (R n ) bounded strong Lpschtz doman Lemma Let Ω be addtonally topologcally trval. Then the followng regular decompostons hold: D( Rot S ) = H 1 S + N( Rot S ), D( Dv T ) = H 1 T + N( Dv T ), D(sym Rot T ) = H 1 T + N(sym Rot T), extenson to general domans... N( Rot S ) = R( Grad grad), N( Dv T ) = R( Rot S ), N(sym Rot T ) = R(dev Grad),

22 RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 Functonal Analyss Toolbox Applcaton: The Rot Rot or Elastcty Complexes Results for Rot Rot or Elastcty Complexes complex D( sym Grad) sym Grad D( Rot Rot S ) Rot Rot S D( Dv S ) Dv S π RM RM adjont complex... general assumpton: Ω R 3 (R n ) bounded strong Lpschtz doman Lemma Smlar results hold.