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Templates for the Solution of Linear Systems

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Templates for the Solution of Linear Systems

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19 Aug 2002 (Updated )

Companion Software

wathen(nx, ny, k)
function A = wathen(nx, ny, k)
%WATHEN  A = WATHEN(NX,NY) is a random N-by-N finite element matrix
%        where N = 3*NX*NY + 2*NX + 2*NY + 1.
%        A is precisely the "consistent mass matrix" for a regular NX-by-NY
%        grid of 8-node (serendipity) elements in 2 space dimensions.
%        A is symmetric positive definite for any (positive) values of 
%        the "density", RHO(NX,NY), which is chosen randomly in this routine.
%        In particular, if D=DIAG(DIAG(A)), then 
%              0.25 <= EIG(INV(D)*A) <= 4.5
%        for any positive integers NX and NY and any densities RHO(NX,NY).
%        This diagonally scaled matrix is returned by WATHEN(NX,NY,1).

%        Reference: A.J.Wathen, Realistic eigenvalue bounds for the Galerkin
%        mass matrix, IMA J. Numer. Anal., 7 (1987), pp. 449-457.

%        BEWARE - this is a sparse matrix and it quickly gets large!

if nargin < 2, error('Two dimensioning arguments must be specified.'), end
if nargin < 3, k = 0; end

e1 = [6,-6,2,-8;-6,32,-6,20;2,-6,6,-6;-8,20,-6,32];
e2 = [3,-8,2,-6;-8,16,-8,20;2,-8,3,-8;-6,20,-8,16];
e = [e1,e2;e2',e1]/45;
n = 3*nx*ny+2*nx+2*ny+1;
A = zeros(n);
rand('uniform')
RHO = 100*rand(nx,ny);

 for j=1:ny
     for i=1:nx

      nn(1) = 3*j*nx+2*i+2*j+1;
      nn(2) = nn(1)-1;
      nn(3) = nn(2)-1;
      nn(4) = (3*j-1)*nx+2*j+i-1;
      nn(5) = 3*(j-1)*nx+2*i+2*j-3;
      nn(6) = nn(5)+1;
      nn(7) = nn(6)+1;
      nn(8) = nn(4)+1;

      em = e*RHO(i,j);

         for krow=1:8
             for kcol=1:8
                 A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol))+em(krow,kcol);
             end
         end

      end
  end

if k == 1
   A = diag(diag(A)) \ A;
end

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