value

2
C. Wieners
c) We solve the linear problem Li v = d i ; where Li is the linearization at ui Li v = ? v + ci v;
v 2 V0 ; ci (x) = @F (x; u (x)): i @y
d) Let ui+1 = ui ? v, i := i + 1 and go to b). The linear problem is solved with a parallel 4-level multigrid V-cycle iteration with 4-colour-block-SOR-smoothing (cf. sec. 2). 1 For the existence proof we consider the nonlinear operator T: H0 ( ) ?! H ?1 ( ) of the form Tu(x) = ? u(x) + F(x; u(x)); x 2 1;1 where F 2 C ( R) such that Tu 2 H ?1 ( ) and Rn is a bounded domain with Lipschitz-continuous boundary. The method is based on the theorem of Newton-Kantorovich. The theoretical aspects are explained e. g. in 8], 11], 16], 17]; the present work describes the numerical aspects. For the application of the existence theorem results we need a defect bound such Z that kT ukH ? ( ) ; ~ where kT ukH ? ( ) := sup ~ T u v dx: ~2 NhomakorabeaZ
ru(x)rv(x) +
F(x; u(x))v(x) dx = 0 8v 2 V0 = fv 2 V j vj@ = 0g;
where : C 2( ) ?! V is a projection and V C 2( ) is a nite element space. We use the Newton method to compute the approximation: a) We start with some u0 2 V0 , i = 0. b) We calculate the defect di = ? ui(x) + F(x; ui(x)) 2 V and stop if kdik ".
1 1
krvkL2(
) =1
In addition, the linearization at u ~
1 L = T 0(~) 2 L(H0 ( ); H ?1( )); Lv = ? v + cv; c(x) = u ~ ~
must be regular and we need a bound such that 1 kvkH ( ) kLvkH ? ( ) 8v 2 H0 ( ); where kvkH ( ) := krvkL ( ) : Both parts, the defect estimation and the norm estimation via eigenvalue enclosures, can be dealt very e ciently with parallel machines (cf. sec. 3). The algorithm is tested for the Bratu problem (cf. 5], 3]) ? u = exp(u) in ; uj@ = 0; > 0: Enclosure results for this problem can be found e. g. in 7], 12], 14], 17]. We improve these results: Using C 2( )-elements we get smoother approximations. The Newton multigrid method converges at every point of the solution branch although the linearization is highly inde nite. Since we have ner grids, we obtain better approximations. This yields a considerable extension of the enclosure results (cf. sec. 4). Most parts of the code consist of local operations. Thus, the parallelization is e ective even for a large number of processors. The algorithm is realized on a 16 16?processor grid of the transputer system (1024 T800 processors) at the university of Cologne.
Z
d e(k;l;i;j ) dx ;
where A is a symmetric n n?matrix (n = 9(2m + 1)2 ). We use a 9 point di erence star to store the matrix: 1 0A A(?1;0;i;j ) A(?1;1;i;j ) (?1;?1;i;j ) A(i;j ) = @ A(0;?1;i;j ) A(0;0;i;j ) A(0;1;i;j ) A ; A(1;?1;i;j ) A(1;0;i;j ) A(1;1;i;j ) where Z A(r;s;i;j ) = (re(k;l;i;j )re(k0;l0 ;i+r;j +s) + c e(k;l;i;j ) e(k0 ;l0 ;i+r;j +s) ) dx 2 R9 9: Analogously, let z(i;j ) = (z(k; l; i; j)) 2 R9. For the computation of A we use tables: let e(k;l;i;j ) 2 P5 be the 36 basis functions ^ (k; l = 0; 1; 2; i; j = 0; 1) of the reference element ^ = (0; 1)2, T1 = and T2 =
Christian Wieners
Mathematisches Institut der Universitat Koln, Weyertal 86-90, 50923 Koln, Germany
Abstract. We describe a parallel algorithm for the numerical computation of guaranteed bounds for solutions of elliptic boundary value equations of second order. We use C 2 -Hermite-elements and a parallel Newton
1 0 1 1 0 2
@F (x; u(x)) @y ~
A parallel Newton multigrid method for high order nite elements
3
2 Multigrid algorithm for high order elements
We describe the multigrid algorithm for = (0; 1)2 and the grid ?m = fxij = (ihm ; jhm ) j i; j = 0; :::; 2mg; hm = 2?m : Let V (m) be the space of C 2( )-Hermite-elements in ?m (cf. 6]). Then, we have vj ij 2 P5 ij = (i ? 1)hm ; ihm ] (j ? 1)hm ; jhm ] for the restriction of v 2 V (m) to a mesh ij , and v is determined by the values ? k l z(k; l; i; j) = hmk?l D1 D2 v(xij ); k; l = 0; 1; 2; i; j = 0; :::; 2m: Let e(k;l;i;j ) 2 V be the corresponding basis functions. Then X v = z(k; l; i; j)e(k;l;i;j ): In every Newton step we solve the linear equation Az = f; f(k; l; i; j) =
1 Introduction
We describe a numerical procedure to solve nonlinear boundary equations of the form ? u(x) + F(x; u(x)) = 0 for x 2 ; uj@ = 0 1 and to prove the existence of a solution u 2 H0 ( ) within error bounds kru ? ru kL ( ) r: ~ Here, u 2 V0 is the approximate solution of the discrete problem ~
multigrid method to produce approximations of high accuracy. Then, we compute upper bounds for the defect and enclosures for the eigenvalues of the linearization. In order to obtain veri ed bounds, these computations are realized in interval arithmetic. The application of the NewtonKantorovich-theorem yields the existence of a solution and error bounds for the approximation. The method is implemented on a 256 processor transputer grid and tested for the Bratu problem ? u = exp(u). AMS Symbol classi cations: 65N55, 65N30, 65Y05, 65N15 Key words: parallel multigrid, nonlinear elliptic boundary equations, error bounds, Bratu problem
A parallel Newton multigrid method for high order nite elements and its application on numerical existence proofs for elliptic boundary value equations