Finite element programming by FreeFem++ advanced course

Transcription

1 1 / 37 Finite element programming by FreeFem++ advanced course Atsushi Suzuki 1 1 Cybermedia Center, Osaka University atsushi.suzuki@cas.cmc.osaka-u.ac.jp Japan SIAM tutorial 4-5, June 2016

2 2 / 37 Outline Basics of FreeFem++ by examples from the Poisson equation Poisson equation with mixed boundary conditions variational forms and stiffness matrix in FreeFem++ Schwarz iterative method overlapping two subdomains and Jacobi-Schwarz method parallel implementation by MPI in FreeFem++ Schwarz algorithm as preconditioner for global Krylov iteration overlapping subdomains and RAS/ASM Krylov subspace method with preconditioner 2-level algorithm with a coarse space Iterative substructuring method / Balancing N-N Schur complement method with preconditioner coarse space constructed from kernel of local problems FETI : Finite element Tearing and Interconnecting interface problem by Lagrange multiplier CG method on the image space with orthogonal projection

3 Numerical simulation with finite element method mathematical modeling discretization of time for evolution problem discretization scheme for the space mesh generation / adaptive mesh refinement stiffness matrix from finite elements and variational formulation linear solver CG, GMRES, direct solver: UMFPACK, MUMPS FreeFem++ provides vast amounts of tools distributed parallelization Krylov subspace method with sophisticated preconditioner by domain decomposition 2-level additive Schwarz method with coarse space iterative method on the interface problem between subdomains iterative substructurinrg method and balancing domain decomposition FETI method 3 / 37

4 4 / 37 Poisson equation with mixed B.C. and a weak formulation Ω R 2, Ω = Γ D Γ N u = f in Ω, u = g on Γ D, u n = h on Γ N. weak formulation function space : V = H 1 (Ω), affine space: V (g) = {u V ; u = g on Γ D }. Find u V (g) s.t. u vdx = Ω a(, ) : V V R : bilinear form F ( ) : V R : functional Ω f vdx + h vds Γ N v V (0)

5 FreeFem++ script to solve Poisson equation finite element basis, span[ϕ 1,..., ϕ N ] = V h V, u h V h u h = 1 i N u iϕ i Dirichlet data : u(p j ) = g(p j ) P j Γ D Find u h V h (g) s.t. u h v h dx = f v h dx + h v h ds v h V h (0). Ω Ω Γ N mesh Th=square(20,20); fespace Vh(Th,P1); Vh uh,vh; func f = 5.0/4.0*pi*pi*sin(pi*x)*sin(pi*y/2.0); func g = sin(pi*x)*sin(pi*y/2.0); func h = (-pi)/2.0 * sin(pi * x); solve poisson(uh,vh)= int2d(th)( dx(uh)*dx(vh)+dy(uh)*dy(vh) ) - int2d(th)( f*vh ) - int1d(th,1)( h *vh ) + on(2,3,4,uh=g); // boundary 1 : (x,0) plot(uh); 5 / 37

6 6 / 37 discretization and matrix formulation FE basis, span[ϕ 1,..., ϕ N ] = V h V, Λ = {1, 2,, N}. u h V h u h = i Λ u iϕ i Dirichlet data : g k = g(p k )k Λ D Λ Find {u j } j Λ s.t. a(ϕ j, ϕ i )u j = F (ϕ i ) i Λ \ Λ D j u k = g k k Λ D mesh Th=square(20,20); fespace Vh(Th,P1); Vh u,v; varf aa(u,v)=int2d(th)( dx(u)*dx(v)+dy(u)*dy(v) ) +on(2,3,4,u=g); // boundary 1 : (x,0) varf external(u,v)=int2d(th)(f*v)+int1d(th,1)(h*v) +on(2,3,4,u=g); real tgv=1.0e+30; matrix A = aa(vh,vh,tgv=tgv,solver=cg); // boundary 1 : (x,0) real[int] ff = external(0,vh,tgv=tgv); u[] = A^-1 * ff; // u : fem unknown, u[] : vector

7 penalty method to solve inhomogeneous Dirichlet problem modification of diagonal entries of A where index k Λ D penalization parameter τ = 1/ε; tgv τ u k = τg k, k Λ D [A] i j = a(ϕ j, ϕ i ) u i f i τu k + a k j u j = τg k u k g k = ε( a k j u j ), j k j k a i j u j = f i i {1,..., N} \ Λ D. j keeping symmetry of the matrix without changing index numbering. 7 / 37

8 8 / 37 alternative Schwarz algorithm (1/2) Ω 2 u 0 1, u0 2 : given loop n = 0, 1, 2, u n+1 1 = f in Ω 1 u n+1 2 = f in Ω 2 u n+1 1 = 0 on Ω 1 Ω u n+1 2 = 0 on Ω 2 Ω u n+1 1 = u n 2 on Ω 1 Ω 2 u n+1 2 = u n+1 1 on Ω 2 Ω 1 essentially sequential interpolation between difference meshes requires Ω 1

9 alternative Schwarz algorithm (2/3) : FreeFem++ script int interface = 2; int original = 1; border a(t=1,2){x=t;y=0;label=original;}; border b(t=0,1){x=2;y=t;label=original;}; border c(t=2,0){x=t;y=1;label=original;}; border d(t=1,0){x= 1-t;y=t;label=interface;}; border e(t=0,pi/2){x= cos(t);y=sin(t);label=interface;}; border e1(t=pi/2,2*pi){x=cos(t);y=sin(t); label=original;}; int n=5; mesh[int] th(2); th[0]=buildmesh(a(5*n)+b(5*n)+c(10*n)+d(5*n)); th[1]=buildmesh(e(5*n)+e1(25*n) ); 9 / 37

10 10 / 37 alternative Schwarz algorithm (3/3) : FreeFem++ script fespace Vh0(th[0],P1); fespace Vh1(th[1],P1); Vh0 u0, v0; Vh1 u1, v1; macro Grad(u) [dx(u),dy(u)] // EOM int i; problem pb0(u0,v0,init=i,solver=umfpack)= int2d(th[0])(grad(u0) *Grad(v0)) -int2d(th[0])(-v0)+on(interface, u0=u1) +on(original,u0=0); problem pb1(u1,v1,init=i,solver=umfpack)= int2d(th[1])(grad(u1) *Grad(v1)) -int2d(th[1])(-v1)+on(interface, u1=u0) +on(original,u1=0); for (i=0 ;i< 10; i++) { pb0; pb1; };

11 11 / 37 Schwarz algorithm : 1/2 Jacobi-Schwarz algorithm u 0 1, u0 2 : given loop n = 0, 1, 2, u n+1 1 = f in Ω 1 u n+1 2 = f in Ω 2 u n+1 1 = 0 on Ω 1 Ω u n+1 2 = 0 on Ω 2 Ω u n+1 1 = u n 2 on Ω 1 Ω 2 u n+1 2 = u n 1 on Ω 2 Ω 1 parallel computation, extendable to more than two subdomains overlapping subdmains

12 12 / 37 Schwarz algorithm : implementation with MPI 1/2 MPI (message passing interface) for distributed parallel computation almost all MPI functions are available in FreeFem++ MPI communicator: mpicomm() broadcasting data from one processor to all others : brodcast() asynchronous data sending/receiving: Isend(),Irecv() synchronization of processes synchronization with message passing requests: mpiwaitany() synchronization of all processes: mpibarrier() reduction operation among processes: mpiallreduce()

13 Schwarz algorithm : implementation with MPI 2/2 mpicomm comm(mpicommworld,0,0); int myrank=mpirank(comm); int orank=(myrank+1)%2; if (myrank==0) th[0]=buildmesh(a(5*n)+b(5*n)+c(10*n)+d(5*n)); else th[1]=buildmesh(e(5*n)+e1(25*n)); broadcast(processor(0,comm),th[0]); broadcast(processor(1,comm),th[1]); mesh tth=th[myrank]; int i; problem pb(u,v,init=i,solver=umfpack) = int2d(tth)(grad(u) *Grad(v)) - int2d(tth)(-v)+on(inside,u=u1)+on(outside,u=0); mpirequest[int] rq(2); for (i=0;i<10; i++) { pb; Irecv(processor(orank,comm,rq[0]),u1[]); Isend(processor(orank,comm,rq[1]),u[]); mpiwaitany(rq); }; 13 / 37

14 Restricted Additive Schwarz algorithm : 1/2 partition of unity u = 2 i=1 E i(χ i u i ) E i : extension from Ω i to Ω χ i : characteristic function in Ω i loop n = 0, 1, 2, = f in Ω i w n+1 i wi n+1 wi n+1 u n+1 = 2 i=1 E i(χ i w n+1 i ) = 0 on Ω i Ω = u n on Ω i Ω j by substituting u n that satisfies u n = 0 on Ω i Ω w n+1 i + u n = f + u n in Ω i wi n+1 u n = 0 on Ω i Ω, wi n+1 u n = 0 on Ω i Ω j u n+1 u n = 2 i=1 E i(χ i w n+1 i ) 2 i=1 E i(χ i u n i ) = 2 i=1 E i(χ i (w n+1 i u n )) 14 / 37

15 15 / 37 Restricted Additive Schwarz algorithm : 2/2 Restricted additive Schwarz (RAS) algorithm u 0 1, u0 2 : given loop n = 0, 1, 2, r n = f + u n vi n+1 = r n in Ω i = 0 on Ω i u n+1 = u n + 2 i=1 v n+1 i E i (χ i v n+1 i ) RAS Jacobi-Schwarz method proof by induction (u n + v n i ) = (u n ) + r n = f in Ω i u n + v n i = u n on Ω i Ω. u n = E 1 (χ 1 u n 1 ) + E 2 (χ 2 u n 2 ) u n = E 1 (0 u n 1 ) + E 2 (1 u n 2 ) = u n 2 on Ω i Ω 2.

16 16 / 37 Additive Schwarz algorithm loop n = 0, 1, 2, wi n+1 wi n+1 wi n+1 u n+1 = 2 i=1 E i(w n+1 i ) = f in Ω i = 0 on Ω i Ω = u n on Ω i Ω j Additive Schwarz Method (ASM) u 0 1, u0 2 : given loop n = 0, 1, 2, r n = f + u n vi n+1 = r n in Ω i = 0 on Ω i u n+1 = u n + 2 i=1 v n+1 i E i (v n+1 i )

17 preconditioned fixed-point iteration cf. V. Dolean, P Jolivet, F. Nataf, An Introduction to Domain Decomposition Methods Algorithms, Theory, and Parallel Implementation, SIAM, ISBN M: preconditioner Ax = b, Mx n+1 = Mx n + (b Ax n ) x n+1 = x n + M 1 (b Ax n ). by setting P = M A, x n+1 = x n + M 1 (b Ax n ) = (I M 1 (M P ))x n + M 1 b = M 1 P x n + M 1 b = x + M 1 P (x n x) x n+1 = x 0 + (M 1 P ) i M 1 (b Ax 0 ) 1 i n x n+1 x 0 span[m 1 r 0, (M 1 P )M 1 r 0, (M 1 P ) n M 1 r 0 ] x n+1 x 0 = s n (M 1 P )r 0 s n (t) = 1 + t + + t n : polynomial Krylov subspace method preconditioned fixed-point iteration 17 / 37

18 conjugate gradient method A u = f. preconditioner Q A 1 Krylov subspace : K n (Q r 0, QA) = span[q r 0, QAQ r 0,..., (QA) n 1 Q r 0 ] Find u n K n (Q r 0, QA) + u 0 s.t. (A u n f, v) = 0 v K n (Q r 0, QA). Preconditioned CG method u 0 : initial step for CG. r 0 = f A u 0 p 0 = Q r 0. loop n = 0, 1,... α n = (Q r n, r n )/(A p n, p n ), u n+1 = u n + α n p n, r n+1 = r n α n A p n, if r n+1 < ɛ exit loop. β n = (Q r n+1, r n+1 )/(Q r n, r n ), p n+1 = Q r n+1 + β n p n. LinearCG(opA,u,f,precon=opQ,nbiter=100,eps=1.0e-10) 18 / 37

19 GMRES method : 1/2 Krylov subspace : K n ( r 0, A) = span[ r 0, A r 0,..., A n 1 r 0 ] Find u n K n ( r 0, A) + u 0 s.t. A u n f A v n f v K n ( r 0, A) + u 0. V m : Arnoldi basis generated by Gram-Schmidt orthogonization for Krylov vectors. u = V m y, y R m J( y) := AV m y r 0 = V T m+1(av m y r 0 ) = (V T m+1av m ) y (V T m+1 r 0 ) = H m y β e 1. (β = r 0 ) Find y R m J( y) J( z) z R m. minimization problem with Hessenberg matrix H m R (m+1) m is solved by Givens rotation. LinearGMRES(opA,u,f,precon=opQ,nbiter=100, eps=1.0e-10) 19 / 37

20 GMRES method : 2/2 Arnoldi method (Gram-Schmidt method on Krylov subspace) v 1 = 1; do j = 1, 2,..., m do i = 1, 2,..., j w j := A v j 1 i j h i j v i, h i j := (A v j, v i ) v j+1 := w j /h j+1 j, h j+1 j := w j Givens rotation matrices Ω i R (m+1) (m+1) Ω i := I i 1 c i s i, c s i c 1 := i Im i Q m := Ω m Ω m 1 Ω 1 R (m+1) (m+1), R m := Q m Hm : upper triangular, ḡ m := Q [ m (βe 1 ) = ] [ γ 1 γ 2 γ m+1 ] T, Rm R m := (R 0 0 m R m m ), ḡ m := h 1 1, s h h 2 1 := 2 2 [ gm γ m+1 h 2 1. h h ] (g m R m ). min βe 1 H m y = min ḡ m R m y = γ m+1 = s 1 s 2 s m β. y m = Rm 1 g m attains the minimum. Remark : Rm 1 (1 m M) for all non-singular matrix A. 20 / 37

21 21 / 37 Schwarz methods as preconditioner ASM preconditioner M 1 M ASM = Rp T (R p ARp T ) 1 R p p=1 ASM does not converge as fixed point iteration, but M 1 ASM is symmetric and works well as a preconditioner for CG method. RAS preconditioner M 1 M RAS = Rp T (R p D p ARp T ) 1 R p p=1 RAS does converge but M 1 RAS is not symmetric and then works as a preconditioner for GMRES method. convergence : slow for many subdomains coarse space

22 22 / 37 2-level Schwarz methods with a coarse space coarse space by Nicolaides D p : discrete representation of the partition of unity M p=1 RT p D p R p = I N, { z p } R N : basis of coarse space, Z = [ z 1,, z M ]. R 0 = Z T. 2-level ASM preconditioner z p = R T p D p R p 1, M 1 ASM,2 = RT 0 (R 0 AR T 0 ) 1 R level RAS preconditioner M 1 RAS,2 = RT 0 (R 0 AR T 0 ) 1 R 0 + M Rp T (R p ARp T ) 1 R p p=1 M Rp T (R p D p ARp T ) 1 R p p=1

23 non-overlapping subdomains and Schur complement method non-overlapping domain decomposition : Ω = Ω 1 Ω 2, Γ = Ω 1 Ω 2, Λ = Λ 1 Λ 2 Λ 3 Λ D : decomposition of DOF, Ω 1, Ω 2, Γ, Ω. A 11 A 13 u 1 f 1 A 22 A 23 u 2 = f 2 + B.C. u = g A 31 A 32 A 33 u 3 f 3 interface problem : S 33 u 3 = (A 33 A 31 A 1 11 A 13 A 32 A 1 22 A 23) u 3 = f 3 A 31 A 1 11 f 1 A 32 A22 1 f 2 subdomain solver: [ A11 A 13 I 3 ] [ ] [ ] w1 0 = w 3 u 3 Dirichlet to Neumann map: u 3 v 3 = (A (1) 33 A 31A 1 11 A 13) u 3 [ ] [ ] [ ] v1 A11 A 13 w1 = v 3 A 13 A (1) w / 37

24 Neumann-Neumann preconditioner S 33 = (A (1) 33 A 31A 1 11 A 13) + (A (2) 33 A 32A 1 22 A 23) Interface problem : Sym. Positive Preconditioned CG Neumann subproblem: [ ] [ ] [ ] A11 A 13 w1 g1 A 13 A (1) = w 33 3 g 3 Note : Neumann problem may be singular in floating subdomain : Ω i Ω =. 1-level Neumann-Neumann preconditioner Q NN = (A (1) 33 A 31A 1 11 A 13) 1 + (A (2) 33 A 32A 1 22 A 23) 1 many subdomains : Λ = Λ I Λ B = (Λ (p) I Λ (p) B ) Q NN = M p=1 RT p D p (A (p) BB A(p BI A(p) II 1 A (p) IB ) D p R p condition nunmer of Q NN S depends on # of subdomains. 24 / 37

25 Balancing Neumann-Neumann preconditioner : 1/3 cf. J. Mandel, Balancing domain decomposition. Commun. Numer. Meth. Engng., 9: DOI: /cnm V : DOF on the skeleton, S : V V, sym. positive definite W : coarse space constructed by vectors Z p in each subdomain including kernel of Neumann problem : [ ] A11 A 13 spanz p Ker A 13 A (1) 33 coarse space on the skeleton M W = { v V ; v = Rp T D p u p, u p spanz p } p=1 P : S-orthogonal projection onto W v = P u : find v W (S( v u), w) = 0 w W 2-level Neumann-Neumann preconditioner Q BNN = (I P )Q NN (I P T ) + P S 1 25 / 37

26 26 / 37 Balancing Neumann-Neumann preconditioner : 2/3 P : S-orthogonal projection onto W coarse grid solver u : given, find v W (S v, w) = (S u, w) find ν q R dimzq Zp T D p R p SRq T D q Z q ν q = Zp T D p R p S u 1 q M p v = 1 q M ZT q D q R q ν q z = Q BNN r find λ q Zp T D p R p SRq T D q Z q λq = Zp T D p R p r p 1 q M s = r S 1 q M RT q D q Z q λq, s p = D p R p s find u p S p u p = s p find µ q Zp T ( D p R p r SR T q D q ( u q + Z q µ q ) ) = 0 1 q M z = 1 q M RT q D q ( u q + Z q µ q ) p

27 Balancing Neumann-Neumann preconditioner : 3/3 z = S 1 r find v W (S( v z ), w) = 0 w W v = P z. s = r S v = S( z v) = S(I P ) z v W u = Q NN s + v find y W (S( z Q NN s ( y + v )), w) = 0 w W y + v = P ( z Q NN s) z = u + y = Q NN s + y + v = Q NN s + P ( z Q NN s) = (I P )Q NN s + P z = (I P )Q NN S(I P ) z + P z = ((I P )Q NN S(I P ) + P ) S 1 r R 0 = [R T 1 D 1Z 1,, R T M D MZ M ] T : restriction operator V W P = R T 0 (R 0 SR T 0 ) 1 R 0 S, P S 1 = R T 0 (R 0 SR T 0 ) 1 R 0, P 2 = R T 0 (R 0 SR T 0 ) 1 R 0 SR T 0 (R 0 SR T 0 ) 1 R 0 S = P, SP = S R T 0 (R 0 SR T 0 ) 1 R 0 S = P T S. 27 / 37

28 28 / 37 Estimation of condition number of BDD Theorem C = sup { 1 q M R q 1 p M RT p D p u p 2 S p 1 p M u p 2 ; S q u p KerS p u p spanz Sp p }. (S u, u) (SM 1 Su, u) C(Su, u) cond(m, S) C. Theorem T pˆx T p ŷ C 1 H ˆx ŷ ˆx, ŷ ˆΩ Tp 1 x Tp 1 y C 2 H 1 x y x, y Ω p Ω p is mapped by T p from a reference square ˆΩ, H = max Ω p. Theorem cond(m, S) c(1 + log 2 H h )

29 p non-overlapping subdomains and Lagrange multiplier : 1/4 cf. C. Farhat, F.-X. Roux., Implicit parallel processing in structural mechanics, Computational Mechanics Advances, 2 (1994) A p u p = f p Bp T λ p Ω B p u p = 0 u p Γ pq = u q Γ (p) Ω (q) pq p u p = A p(f p Bp T λ) + R p α p, span R p = KerA p [ ] [ ] [ A11 A A p = 12, A A 1 p = 11 0 A 1, R A 21 A p = 11 A ] 12 I 2 compatibility condition for external force: R T p (A p u p ) = R T p (f p B T p λ) = 0 continuity on the interface: ( Bp A p(f p Bp T ) λ) + B p R p α p = 0 29 / 37 λ

30 30 / 37 non-overlapping subdomains and Lagrange multiplier : 2/4 p B p A pb T p λ + p B p R p α p = p B p A pf p R T p B T p λ = R T p f p by setting F = p B pa pb T p, G = [B 1 R 1, B 2 R 2,, B M R M ] [ ] [ ] F G λ G T = 0 {α p } p [ fλ f α ] f λ = p B pa pf p f α, p = R T p f p F : positive semi-definite due to cross point of decomposition originally non-floating structure G has full column rank proof: assume that G does not have full column rank γ = {γ p } 0 s.t. p B P R p γ p = 0. Ω p Ω, 0 Dirichlet b.c. R p γ p = 0 then R q γ q = 0 q.

31 non-overlapping subdomains and Lagrange multiplier : 3/4 P : orthogonal projection onto KerG T, P = I G(G T G) 1 G T. reformulating interface problem of Lagrange multiplier on KerG: find λ : [ ] [ ] [ ] F G λ + λ fλ = G T 0 [ F G G T 0 f α [ fλ α ] [ ] ] [ ] [ ] λ F G λ = α f α G T 0 0 [ ] [ ] fλ F λ = fλ F λ f α + G T = λ 0 λ = G(G T G) 1 [f T 1 R 1,, f T M R M] T. CG method on KerG T by the orthogonal projection P, P F λ = P (f λ F λ ). F λ = (f λ F λ ) + Gα α = (G T G) 1 G T (F (λ + λ ) f λ ). finally, u p = A p(f p B T p (λ + λ )) + R p α p = ū p + R p α p = ū p R p [(G T G) 1 G T q B qū q ] p. 31 / 37

32 conjugate gradient method with orthogonal projection orthogonal projection P : R N KerG T find u KerG T P A u = P f u 0 KerG T, r 0 = P f P A u 0 KerG T Krylov subspace : K n ( r 0, P A) = span[ r 0, P A r 0,..., (P A) n r 0 ] Find u n K n ( r 0, P A) + u 0 s.t. (A u n f, v) = 0 v K n ( r 0, P A). Projected CG method u 0 KerG T : initial step for CG. r 0 = P ( f A u 0 ) p 0 = r 0. loop n = 0, 1,... α n = ( r n, r n )/(P A p n, p n ), u n+1 = u n + α n p n, r n+1 = r n α n P A p n, if r n+1 < ɛ exit loop. β n = ( r n+1, r n+1 )/( r n, r n ), p n+1 = r n+1 + β n p n. 32 / 37

33 non-overlapping subdomains and Lagrange multiplier : 4/4 F = p B pa pb T p is positive semi-definite cross point, e.g. Q; B q u q (Q) + B p u p (Q) = 0 B p u p (Q) + B r u r (Q) = 0 B r u r (Q) + B s u s (Q) = 0 B s u s (Q) + B q u q (Q) = 0 removing redundant DOF of Lagrange multiplier: Ω (r) Ω (s) Q Ω (p) Ω (q) λ f λ = p B pa pf p ImF guarantees convergence of CG in ImF. 33 / 37

34 preconditioned CG method with orthogonal projection : 1/2 orthogonal projection P : R N KerG T find u KerG T P A u = P f, u 0 KerG T, r 0 = P f P A u 0 Q : preconditioner, Q A 1. Krylov subspace : K n ((P Q) r 0, P QP A) = span[(p Q) r 0, (P QP A)(P Q) r 0,..., (P QP A) n (P Q) r 0 ] Find u n K n ((P Q) r 0, P QP A) + u 0 s.t. (A u n f, v) = 0 v K n ((P Q) r 0, P QP A). Projected PCG method u 0 KerG T : initial step for CG. r 0 = P ( f A u 0 ) p 0 = P Q r 0. loop n = 0, 1,... α n = (P Q r n, r n )/(P A p n, p n ), u n+1 = u n + α n p n, r n+1 = r n α n P A p n, if r n+1 < ɛ exit loop. β n = (P Q r n+1, r n+1 )/(P Q r n, r n ), p n+1 = P Q r n+1 + β n p n. 34 / 37

35 35 / 37 preconditioned CG method with orthogonal projection : 2/2 Dirichlet preconditioner Q D = [ ] 0 0 B p Bp T, 0 S bb p S bb = A bb A bi A 1 ii A ib lumped preconditioner Q L = p [ ] 0 0 B p Bp T 0 A bb FETI method converges fast when the domain is cut with smooth interafce segments. Dirichlet preconditioner is better than lumped one but two local solvers are necessary.

36 36 / 37 extension of FreeFem++ by dynamic loading capability dynamic loading functions are in examples++-load source is written by C++ and called from edp script by using load "PARDISO" etc. several capabilities are extended without changing the original FreeFem++ sources linear solver : PARDISO, MUMPS mesh decomposer : metis, scotch finite elements : Element_P3, Element_Mixte output : VTK_writer The simplest examples in examples++-load myfunction.cpp and PARDISO.cpp

37 PARDISO in MKL by Intel void pardiso(_mkl_dss_handle_t pt,// pardiso handler MKL_INT *maxfct, // #of factorization(=1) MKL_INT *mnum, // 1<=mnum<=maxfct (=1) MKL_INT *mtype, // MKL_INT *phase, // MKL_INT *n, // size of matrix void *a, // coefficients MKL_INT *ia, // pointer where i-th row MKL_INT *ja, // column index of CSR MKL_INT *perm, MKL_INT *nrhs, // #of RHS vectors MKL_INT *iparm, MKL_INT *msglv, // message level void *b, // RHS void *x, // solution MKL_INT *error); mtype real structurally symm. SPD sym. indefinite phase symbolic factorization symbolic + numeric fw/bw solve 37 / 37